LIBRARY 


University  of  California. 


Class 


\ 


QUATERNIONS 


AS  THE 


RESULT  OF  ALGEBHAIC  OPERATIONS 


BY 


ARTHUR   LATHAM   BAKER,   Ph.D. 

Head  of  Department  of  Mathematics,  Manual  Training  High  School, 
Brooklyn,  N.  Y. 


NEW    YORK 

D.  VAN    NOSTRAND    COMPANY 

23  MURRAY  AND  27  WARREN  STREETS 
1911 


T53 


Copyright,  1911 

BY 

D.   VAN    NOSTRAND    COMPANY 


THE  SCIENTIFIC   PRESS 

ROBERT   DRUMMOND   AND   COMPANY 

BROOKLYN,    N.   Y. 


PREFACE 


Beginners  in  the  subject  of  Quaternions  are  generally 
bewildered  by  the  arbitrary  manner  in  which  the  subject 
is  developed.  They  are  forcibly  introduced  into  a  new 
domain  where  the  familiar  rules  of  combination  of  symbols 
are  not  valid.  New  magnitudes  are  arbitrarily  assumed, 
subject  to  arbitrary  laws.  The  reader  finds  the  logic  con- 
sistent and  the  results  concordant  with  those  of  his  previous 
courses,  but  he  hardly  knows  why.  He  finds  himself  in  a 
new  country,  but  thoroughly  and  bewilderingly  uncertain 
as  to  how  he  got  there. 

It  is  in  the  attempt  to  avoid  this  uncertain  journey,  to 
lead  the  student  from  the  known  to  the  unknown  by 
familiar  steps,  by  steps  which  require  no  arbitrary  limita- 
tions of  former  laws,  but  merely  their  adaptation  to  new 
circumstances,  that  these  class  notes  have  grown  into  their 
present  shape. 

The  backbone  of  the  method  of  presentation  is  the  use 
of  a  one-to-one  correspondence  between  the  mathematical 
concept  and  what  I  have  ventured  to  call  its  idiographic 
symbol,  that  is,  a  symbol  wh^e  spatial  properties  are  the 
same  as  the  mathematical  properties  of  the  concept  it 
symbolizes.  From  this  similarity  of  properties  there  exists 
a  one-to-one  correspondence  between  the  results  of  spatial 
operations  upon  the  symbols  and  the  corresponding  mathe- 
matical operations  upon  the  concept. 

iii 


236365 


iv  PREFACE 

These  idiographic  symbols  are  strokes,  spherical  shells, 
and  vectors,  corresponding  respectively  to  magnitudes 
having  size,  and  correlated  sense  of  opposition,  scalars,  and 
magnitudes  having  size,  sense,  and  direction. 

Spatial  operations  upon  these  symbols  are  used  as  sug- 
gestions for  a  one-tc-one  corresponding  interpretation  for 
for  the  mathematical  concept. 

These  spatial  operations  are  rational  and  logical  and  require 
no  "  standing  loose  for  a  time  to  logical  accuracy."  *  As 
they  are  rational  and  logical,  so  their  interpretations  are 
rational  and  logical,  and  the  reader  does  not  lose  his  sense 
of  logical  sequence.  There  is  no  "  removal  of  barriers,  of 
limitations,  of  conditions."  *  Multiplication  is  the  same 
from  beginning  to  end,  whether  applied  to  scalars,  vectors,  or 
quaternions.  Commutivity  of  factors  may  be  permissible 
in  some  cases  and  not  in  others,  but  this  is  a  mere  incident 
and  not  an  essential  element  of  the  operation. 

The  reader  is  not  mystified  by  arbitrarily  defining  multi- 
plication of  one  vector  into  another  as  the  turning  through 
a  right  angle,  etc.,  and  left  to  wonder  how  one  line  can  do 
anything  to  another.  In  fact,  the  operations  are  not 
defined  a  priori  at  all,  but  taking  the  properties  of  discrete 
quantities  as  symbols  of  operations  which  the  reader  is  to 
perform,  we  find  six  possible  operations,  addition,  subtrac- 
tion, multiplication,  division,  reversion,  and  mean  reversion. 
These  operations  are  defined  accordingly  a  posteriori  as 
results  of  causes,  not  arbitrarily  as  assumptions. 

The  performance  of  these  operations  upon  scalars  leads  to 
or  evolves  successively  vectors  and  quaternions.  Thus  qua- 
ternions are  evolved  from  discrete  magnitudes,  not  arbitra- 
rily, but  of  necessity,  and  along  certain  fixed  and  preordained 

*  Kelland  and  Tait,  and  others. 


PREFACE  V 

lines,  by  rules  which  the  properties  of  discrete  magnitudes 
necessitate,  and  which  cannot  be  altered  or  varied,  and 
with  which  the  reader  is  already  familiar.  The  reader  is 
not  disturbed  by  the  thought.  Suppose  we  had  made  some 
other  assumption,  what  then?  No  assumptions  are  made. 
He  simply  follows  the  road  suggested  by  the  properties  of 
discrete  magnitudes,  and  can  arrive  at  but  one  result. 

We  make  no  laws,  lay  down  no  rules,  make  no  modifica- 
tions or  limitations.  The  only  way  in  which  we  exercise 
any  choice  is  in  the  rational  application  of  the  laws  we 
discover  to  the  proper  operands  and  in  a  proper  and  logical 
manner. 

The  ''  interpretation  of  our  results "  is  not  made  to 
"  depend  upon  the  definition  "  as  a  foundation.  The  foun- 
dation is  the  properties  of  discrete  magnitudes,  and  the 
definitions  are  merely  rational  statements  of  the  results  of 
these  properties  being  used  as  suggestions  for  operations  to 
be  performed  by  the  reader. 

Considerable  stress  has  been  laid  upon  the  avoidance  of 
the  sole  use  of  mere  typographical  symbols  and  upon  the 
auxiliary  use  of  idiographic  symbols  upon  which  spatial 
operations  can  be  performed;  as  in  the  use  of  strokes  for 
merely  reversible  magnitudes,  a  spherical  shell  for  scalar 
magnitudes,  arc  strokes  for  quaternion  multiplication,  two 
vector  factors  for  the  corresponding  quaternion,  etc.,  thus 
making  the  treatment  concrete  and  avoiding  the  difficulties 
of  abstractness. 

The  original  features  of  the  book  are  those  specified  above, 
coupled  with  the  general  heuristic  method  by  which  the 
student  hews  out  his  own  concepts  as  he  goes  along.  The 
results,  the  examples,  the  applications,  and  the  terms  used 
are  those  found  in  every  treatise  on  the  subject,  of  which 
I  have  made  free  use  and  to  whom  should  be  accredited 


vi  PREFACE 

these  features:  particularly  Hamilton,  Kelland  and  Tait, 
Tait,  Laisant,  Molenbrock,  Hathaway. 

As  these  notes  are  only  intended  as  an  introduction,  not 
an  overabundance  of  examples  or  formula?  has  been  pro- 
vided, nor  have  any  applications  been  made  to  problems  in 
Geometry  and  Physics.  These  will  be  found  in  the  works 
cited.  Nor  has  the  subject  of  differentiation  been  touched 
upon. 

The  author's  own  experience  with  this  method  of  presenta- 
tion of  the  subject  to  beginners  has  been  encouraging.  It 
is  hoped  others  will  have  the  same  experience. 


CONTENTS 


CHAPTER  I 

PAGE 

Mathematical  Operations  upon  Discrete  Magnitudes 1 

Discrete  magnitudes,  1;  Addition,  4;  reversion,  5;  sub- 
traction, 6;  multiplication,  7;  division,  8;  mean  reversion,  9. 

CHAPTER  II 

Idiographs 6 

Strokes,  11;  Argand  diagram,  13;  addition  of  strokes,  15; 
algebraic  operations,  17;  scalars,  18. 

CHAPTER  III 

Space  Idiographs 10 

Space  idiographs,  19;  mean  reversion,  20;  vector,  22; 
reversal,  26;  addition  and  subtraction  of  vectors,  27; 
decomposition  of  vectors,  29;  notation,  30;  exercises  in 
combination,   31;  examples,  37. 

CHAPTER  IV 

Multiplication  of  Unit  Vectors 17 

Parallel  vectors,  40;  perpendicular  vectors,  41;  inclined 
vectors,  42;  exercises  in  i,  j,  k,  44. 

vii 


vm  CONTENTS 

CHAPTER  V 

PAGE 

Quaternions 21 

Quaternion,  45;  meaning  of  — ,  47;   examples,  48;   commu- 

tivity  of  vectors,  49;  types  of  quaternions,  50;  notation,  53; 
exercises,  54;  geometric  meaning  of  TVafi  and  Sa^,  58; 
meaning  of  Sa^Y,  59;  sum  of  scalar  and  vector,  61;  definitions 
of  algebra,  etc.,  63. 

CHAPTER  VT 

Kinds  of  Quaternions 32 

Reciprocal  of  a  quaternion,  64;  opposite  quaternions,  65; 
conjugate  quaternions,  66;  table  of  quaternions,  69;  equality 
of  quaternions,  71;    diplanar  quaternions,  72. 


CHAPTER  VII 

Quaternion  Operators 37 

Quaternion  as  a  multiplier,  73;  angle  of  a  quaternion,  76; 
arc  strokes,  77;  multiplication  and  division  by  arc  strokes,  80; 
table  showing  /g,  82;  arc  stroke  of  a  vector,  84;  table,  85; 
square  root  by  strokes,  86;  examples,  88. 


CHAPTER  VIII 

Products  of  Quaternions 45 

Productsin  general,  90;  conjugateof  a  product,  91;  coplanar 
quaternions,  92;  examples,  93. 


CHAPTER  IX 

Versors 48 

Versors,  94;  multiplication  of  versors,  97;  examples,  99; 
conversion  of  qr  into  rq,  101;  meaning  of  qrq~~^,  102;  examples, 
103. 


CONTENTS  ix 

CHAPTER  X 

PAGE 

Interpretation  of  Vector  Equations 54 

a^y  =  H,   107;     exercises  and  formulae,    111;    applications, 
114;    reference  formulae,  122. 


CHAPTER  XI 

Quaternion  Equation  of  the  First  Degree 62 

General  form,  127;  linear  scalar  equation,  129;  decomposi- 
tion of  vector  part,  130;  linear  vector  function,  131;  properties 
of  0,  132;  conjugate  strain  function,  135;  application  of  <p 
to  vector,  141;  properties  of  (J),  143;  nonion,  145. 


CHAPTER  XII 

Applications  of  ^ 71 

Change  of  volume,  146;  special  values  of  modulus,  148; 
special  applications  of  cfi,  153;  properties  of  (0'  —  g),  154;  latent 
lines,  latent  roots,  latent  planes,  162;  two  latent  roots  equal, 
164;  three  latent  roots  equal,  165;  pure  strain,  172;  examples, 
175. 

Appendix 87 

Functional  Symbols. 


QUATERNIONS 


CHAPTER   I 


MATHEMATICAL  OPERATIONS  UPON  DISCRETE 
MAGNITUDES 

1.  Mathematics  is  for  school  purposes  the  science  of 
magnitude. 

Magnitude  has  been  defined  as  that  which  can  be  increased 
or  diminished,  and  of  two  kinds,  continuous  magnitude 
(continuum),  which  is  usually  called  quantity  and  answers 
to  the  question,  How  much?  and  discrete  magnitude 
(discreta),  which  is  usually  called  number  and  answers  to 
the  question.  How  many? 

Quantity  is  differentness  from  nothingness,  or  quantita- 
tively expressed,  difference  from  nothing. 

2.  When  we  attempt  to  express  quantity  in  symbols  or 
terms,  or  to  conceptualize  it,  we  find  ourselves  compelled 
to  express  it  in  terms  of  some  arbitrary  unit  adopted  as  a 
standard.  The  symbol  or  concept  for  the  measure  of  quan- 
tity is  called  number,  expressing  its  differentness  from 
oneness,  the  oneness  of  the  standard. 

In  the  case  of  natural  units,  soul,  personality,  inhabitant, 
etc.,  the  differentness  from  oneness  which  distinguishes  a 
group  from  an  individual  is  number,  giving  rise  to  the 
so-called  primary  numbers,  1,  2,  3,  .  .  . 

Number    is    differentness    from     oneness.     The    whole 


2  QtrA-TERNIONS 

essence  of  number  is  to  differ  from  another  number.  It 
belongs  to  the  primary  categories  of  time,  space,  matter, 
etc.  No  one  of  these  can  exist  without  the  presence  of 
number,  that  is  without  a  oneness  and  a  difference  from 
that  oneness.  The  very  existence  of  time,  space,  matter 
imphes  multipHcity ,  three-f oldness. 

The  Hmitations  of  the  mind  require  symbols  to  represent 
the  different  concepts  so  that  the  mind  can  posit  its  judg- 
ments and  conclusions  to  await  its  return  to  make  use  of 
them.  Otherwise  the  mind  soon  becomes  clogged  with  the 
impedimenta  of  its  own  creation. 

Upon  the  fortunate  choice  of  these  symbols  depends  very 
greatly  the  progress  in  their  use.  A  striking  illustration  of 
this  is  seen  in  the  enormous  advantage  of  the  Hindu  position 
system  over  the  clumsy  notation  of  the  Greeks  and  Romans. 

3.  In  the  infancy  of  the  subject  pebbles  (calculi)  and 
other  concrete  objects  were  the  symbols  used.  And  as  a 
negative  pebble  was  impossible  of  imagination,  negative 
numbers  were  imaginary.  Even  as  late  as  the  sixteenth 
century  they  were  called  by  Cardan  in  his  Ars  Magna, 
numeri  ficti,  imaginary  numbers.  After  they  were  dignified 
with  the  title  of  real  numbers,  expressions  containing  \/^^ 
would  intrude  themselves,  under  the  operations  of  mathe- 
matical analysis,  and  as  there  was  no  way  of  writing  a 
numerical  symbol  whose  square  would  be  a  negative 
number,  these  were  in  their  turn  imaginaries,  even  to  the 
present  day.  So  late  a  writer  as  De  Morgan  (1831)  speaks 
of  them  as  void  of  meaning,  self-contradictory,  and  absurd, 
though  of  great  utility  in  the  formal  mechanism  of  Algebra. 

But  it  is  not  the  number  that  is  imaginary,  it  is  the  appli- 
cation of  it  to  certain  data  or  symbols  that  produces  the 
imaginary  features.  Fractions  are  imaginary  when  count- 
ing souls,  inhabitants,  events,  phenomena,  etc.     Negative 


MATHEMATICS  AND   DISCRETE   MAGNITUDES  3 

numbers  are  imaginary  when  applied  to  length,  bulk, 
intensity,  etc.  The  imaginaries  of  to-day  become  real 
when  applied  to  the  proper  symbol. 

Before  searching  for  a  symbol  we  must  inquire  about  the 
properties  of  primary  number  and  the  operations  perform- 
able  upon  it.  What  are  these  performable  operations, 
guiding  ourselves  by  the  properties  of  number  itself?  This 
guidance  is  necessary,  the  properties  of  a  thing,  if  we  thus 
speak  of  number,  being  the  determining  factors  as  to  what 
purposes  it  can  be  put. 

4.  Number  is  differentness  from  unity.  We  can  increase 
this  differentness  quantitatively  by  combining  the  different- 
ness of  one  number  with  the  differentness  of  another 
number.  This  operation  we  call  addition,  and  by  means 
of  it  we  can  evolve  all  the  primary  numbers  from  unity. 
In  this  two  numbers  are  taken  accumulatively. 

5.  We  can  imagine  the  differentness  from  oneness  to  have 
a  correlated  sense  of  opposition,  like  debit  and  credit, 
affirmation  and  negation,  an  antagonistic  differentness. 
One  is  the  reversal  of  the  other,  the  negative  of  the  other; 
and  the  process  of  converting  one  into  the  other  is  called 
reversion. 

Numbers  having  this  correlated  sense  of  opposition  are 
said  to  have  size  and  sense.  An  example  could  be  the 
accumulative  collection  of  material  into  the  form  of  a  mound 
or  tumulus.  The  workmen  could  proceed  decumulatively 
(this  word  is  not  in  the  lexicons,  but  its  usefulness  is  apparent) 
by  undoing  their  former  work,  and  might  proceed  below 
the  bed  rock.  We  might  not  call  this  a  negative  tumulus, 
but  it  would  be  the  negative  of  the  tumulus  concept  and 
the  number  concept  would  be  a  negative  number. 

6.  We  can  diminish  the  differentness  from  oneness  which 
is  characteristic  of  a  number  by  the  characteristic  different- 


4  QUATERNIONS 

ness  of  some  other  number,  giving  rise  to  the  operation  called 
subtraction.  The  continued  application  of  this  process  to 
numbers  having  size  and  sense  leads  to  negative  numbers. 
Here  the  numbers  are  taken  decumulatively.  As  addition 
is  accumulative  combination  so  subtraction  is  decumula- 
tive  combination  of  numbers  without  regard  to  the  size  of 
the  subtrahend. 

We  can  consider  the  characteristic  differentness  of  one 
number  as  a  mandate  for  an  operation  to  be  performed 
upon  another  number.  Now  this  characteristic  different' 
ness  from  oneness  or  unity  may  be  evolutory  or  involutory, 
differentness  from  unity,  .or  differentness  toward  unity,  a 
differentness  which  evolves  the  number  from  unity,  or  a 
differentness  which  converts  a  number  into  unity. 

7.  If  we  take  the  evolutory  differentness  as  the  mandate, 
then  the  operation  is  called  multiplication,  the  doing  to 
the  operand  (multiplicand)  what  was  done  to  unity  to 
produce  the  operator  (multiplier).  If  the  first  operand  and 
all  the  operators  in  a  series  of  operations  are  the  same,  the 
operation  is  called  involution.  Involution  works  from  unity 
with  the  first  operand  and  all  the  operators  alike. 

A  curious  error  is  liable  to  creep  in  here  unless  care  be 
exercised.  For  example,  in  V2-3,  one  is  liable  to  say  that 
\/2  is  derived  from  1  by  doubling  1  and  taking  the  \/~  of 
the  result.  This  applied  to  3  gives  the  result  \  2T3,  which 
is,  of  course,  wrong.  The  error  consists  in  taking  the 
square  root  of  the  result,  whereas  multiplication  is  the 
doing  what  was  done  to  unity,  and  not  what  was  done  to 
the  results  of  operations  upon  unity.  To  get  \/2  from 
unity,  we  really  operate  upon  unity  with  the  tensor  or 
stretcher  1.4142  ..  .  =\/2. 

8.  If  we  take  the  differentness  toward  unity  as  the  man- 
date, then  the  operation  is  called  division,  the  doing  to  the 


MATHEMATICS   AND  DISCRETE  MAGNITUDES  5 

operand  (dividend)  what  was  done  to  the  operator  (divisor) 
to  produce  unity. 

If  all  the  operators  and  the  last  operand  in  a  series  of 
operations  are  the  same,  the  operation  is  called  evolution, 
and  gives  rise  to  irrational  and  so-called  imaginary  numbers, 
e.g.,  V2,  v-T,  etc. 

Evolution  works  towards  unity  with  all  the  operators  and 
the  last  operand  alike. 

9.  One  of  the  most  important  phases  of  evolution  is  \/a^ 
the  breaking  up  of  the  operation  of  passing  from  unity  to 
a  into  two  equal  steps,  so  that  the  repetition  of  the  first 
step,  \/a,  shall  produce  a.     When  a  =  —  1,  we  get  V^^. 

Since  —  1  is  reversion,  \/^^  is  called  mean  reversion. 

Unlike  addition  and  subtraction,  which  merely  adjoin  the 
elements  without  change,  multiplication  and  division  are 
transformational  operations  which  change  the  operand  into 
an  entirely  new  and  different  number.  They  are  purely 
algebraic,  being  limited,  unlike  addition  and  subtraction, 
to  operations  upon  discrete  number.  A  not  entirely  satis- 
factory illustration  of  the  difference  between  the  two 
operations  of  addition  and  multiplication  is  the  putting  of 
two  canes  together  for  addition;  for  multiplication  we 
would  be  compelled  because  the  first  was  triangular,  black, 
smooth,  and  uniform  in  size  to  transform  the  second  cane 
which  was  round,  white,  knotted,  tapering,  and  with  a 
knob  on  the  end,  into  a  triangular,  black,  smooth,  prismatic 
stick.  It  is  transformed  into  an  entirely  new  and  distinct 
form,  and  its  original  features  have  disappeared.  So  in 
numbers,  addition  and  subtraction  are  merely  adjunction 
without  change  of  the  original  elements.  Multiplication 
and  division  are  destructive  transformations,  the  destruction 
of  one  element  and  the  production  of  an  entirely  new  and 
different  one,  like  the  growth  of  a  plant  from  a  seed. 


CHAPTER  II 
IDIOGRAPHS 

10.  Mathematical  operations  are  conducted  by  means  of 
symbols,  and  upon  our  choice  of  symbols  depends  Largely 
the  success  of  our  operations. 

In  the  dawn  of  mathematics,  the  fingers  or  other  concrete 
objects  were  the  symbols  used,  and  a  negative  quantity  was 
purely  imaginary.  With  the  introduction  of  written 
symbols  of  quantity  and  the  concept  of  debit  and  credit, 
the  negative  quantity  lost  its  imaginary  quality  and  became 
real.  In  these  symbols,  however,  V^^  remained  imagi- 
nary, because  the  spatial  and  typographical  properties  of 
the  symbol  and  the  mathematical  properties  of  the  thing 
it  represented  did  not  agree. 

11.  With  the  introduction  of  the  symbol  called  a  stroke, 
a  straight  line  upon  the  surface  of  the  paper,  to  represent  a 
magnitude  of  a  given  size,  the  space  properties  of  the 
symbol  and  the  mathematical  properties  of  the  thing 
symbolized  became  the  same,  and  spatial  operations  upon 
the  symbols  corresponded  to  mathematical  operations  upon 
the  thing  symbohzed,  and  could  be  used  to  interpret  and 
control  the  mathematical  operations. 

Symbols  whose  spatial  properties  are  the  same  as  the 
mathematical  properties  of  the  things  represented  might 
be  called  idiographs  {iBios,  proper,  peculiar) . 

12.  Representing  a  magnitude  having  sense  .of  generation, 
sense  of  correlated  opposition,  by  the  idiograph >,  and 

6 


i/T 


IDIOGRAPHS  7 

of  course  its  reversal  by  < ,  how  can  we  convert >  into 

^ ? 

Obviously,  only  in  one  way,  so  long  as  we  retain  its 
stroke  characteristics,  namely,  by  swinging  it  through  an 
angle  of  180°  in  the  surface  of  the  paper.  To  attempt  to 
swing  it  out  of  the  surface  of  the  paper  is  to  lose  its  stroke 
characteristics,  to  give  it  absolute  direction  in  space  and 
render  it  no  longer  an  idiograph. 

The  revolution  of  180°  can  be  broken  into  two  equal  steps 
of  90°  each.  Hence  |  or  |  must  be  the  idiographic 
equivalent  of  \/~- 

Combining   these   into  one  diagram   and  i/^ 

assuming  the  normal  revolution  as  counter-       _j 
clockwise,  we  get  the  idiographic  diagram 
with  the  corresponding  typographical  sym- 
bols. 

13.  This  is  the  well-known  Argand  diagram,  affording  a 
simple  method  of  representing  relatively  directed  quantities, 
or  as  they  are  generally  called,  complex  quantities,  the  general 
type  of  which  is  x+a/^^?/,  where  x  represents  the  normal 
o'r   the   reversed   portion   and   ^y  —  ly   the   mean   reversed 

portion.  It  is  generally  written 
x-\-iy,  i  standing  for  \/^^.  The 
rectangular  co-ordinates  x  and  iij 
determine  a  point,  whose  distance 
from  the  origin,  r,  is  called  the  mod- 
ulus,   and    whose    angular    distance 

from  the  axis  of  x,  the  angle  (j),  is  called  the  amplitude 

of  the  point  or  complex  quantity  x+iy. 

14.  The  idiographic  symbol  for  a  magnitude, >,  show- 
ing its  size  and  sense,  we  have  already  designated  as  a 
stroke,  a  stroke  forward  or  a  stroke  backward.  Two 
forward    strokes   need   not   be   represented   by   the    same 


8  QUATERNIONS 

symbol  on  the  paper.  The  forwardness  is  in  reference  to 
its  own  backwardness,  and  has  no  reference  to  the  forward- 
ness or  backwardness  of  other  strokes.  A  stroke  is  a 
straight  Kne  in  a  plane,  symbolizing  a  given  magnitude  in 
size  and  in  relation  to  its  sense  of  normalcy  or  of  reversion, 
or  a  condition  betw^een  these.  Two  strokes  are  equal  when 
they  have  the  same  lengths  and  the  same  direction  in  a 
plane  as  regards  a  standard  normal  direction  in  the 
plane. 

15.  If  several  strokes  be  taken  in  succession  the  sum 
(result)  of  them  is  the  same  as  the  stroke  from  the  beginning 
of  the  first,  to  the  end  of  the  last,  when  they  are  arranged 
end  to  end  so  as  to  be  successive.     Thus 

Strokes  will  for  the  present  be 
represented  typographically  by  lower- 
carse  Greek  letters,  as  above. 

The  reader  must  notice  carefully 
that  it  is  not  the  lengths  of  the  stroke 
which  are  added,  but  the  results  of 
the  strokes,  including  both  length  and  direction  on  the 
paper. 

It  is  easily  seen  that  the  order  of  the  strokes  is  immaterial, 
and  that  any  number  of  consecutive  strokes  can  be  replaced 
by  their  sum.  The  addition  of  strokes  is  a  commutative  and 
associative  operation,  that  is,  the  order  and  mode  of  grouping 
has  no  effect  on  the  result. 

A  stroke  is  subtracted  by  reversing  its  direction  and 
adding. 

16.  If  we  attempt  to  break  up  the  operation  of  rever- 
sal into  three  or  more  equal  and  similar  operations, 
for  example,  three,   a^   shown    in    the   diagram,    we   find 


IDIOGRAPHS  9 
1         -^/q" 

that  ^'  — 1=— -f-t; ,  that  is,  the  first  of   the  three  equal 

operations    is    expressible   in   terms    of    the   operation   of 
mean  reversion,  and  of  course  the  second 
operation  likewise. 

Similarly    for    a    greater    number    of 

steps.     Hence    mean    reversion    is   evi-   ' ^— ^ 

dently  the  unique  operation,  a  multiple  of  which  is  rever- 
sion, all  the  other  partial  equal  operations  whose  con- 
tinued application  results  in  reversion  being  expressible 
in  this  one. 

17.  Hence  there  are  six  unique  and  fundamental  opera- 
tions which  can  be  performed  upon  a  discrete  magnitude: 
addition,  subtraction,  reversal,  multiplication,  division,  and 
mean  reversion,  and  no  others. 

By  fundamental  operations  is  meant  operations  based 
upon  the  properties  of  discrete  magnitudes,  size,  and  sense 
of  correlated  opposition. 

18.  Mere  discrete  magnitudes,  considering  size  only,  are 
scalars,  that  is,  they  can  be  scaled  off  on  a  scale  either  in 
a  normal  or  in  a  reversed  direction. 


CHAPTER   III 

SPACE   IDIOGRAPHS 

19.  Space  idiographs.  In  space  we  cannot  idiograph- 
ically  represent  a  scalar  by  a  line,  for  that  would  be  assign- 
ing to  the  symbol  a  characteristic  direction,  which  the 
magnitude  it  represents  does  not  possess.  If  we  are  to 
represent  a  scalar  magnitude  in  space  by  any  idiographic 
symbol,  the  only  one  which  seems  available  as  possessing 
perfect  symmetry  and  therefore  devoid  of  direction  is  a 
spherical  shell. 

Just  as  we  can  assume  our  +  unit  of  heat,  pressure,  etc., 
anywhere  on  the  scale,  so  we  can  posit  our  +  spherical  shell 
anywhere  in  space. 

Likewise  as  the  —  unit  of  heat,  etc.,  would  naturally 
adjoin  the  +  unit,  along  the  scale,  so  naturally  we  should 
expect  the  —  unit  shell  to  adjoin  the  +  unit  shell  in  some 
position  determined  by  previous  assignment  of  direction. 
According  to  this  previously  determined  direction  we  shall 
have  units  of  a  direction,  of  /?  direction,  etc.,  where  a  and 
/?  denote  direction,  not  magnitudes,  unless  we  say  unit 
magnitude,  just  as  previously  we  had  units  of  heat  sense, 
credit  sense,  etc. 

20.  How  can  we  break  up  the  operation  of  transforming 
the  +  shell  into  the  —  shell  into  two  similar  and  equal 
operations.  The  +  shell  can  be  changed  into  the  —  shell 
by  the  repetition  of  two  different  operations: 

10 


SPACE  IDIOGRAPHS  11 

A.  By  revolving  the  —  shell  about  its  point  of  contact 
with  the  +  shell  through  an  angle  of  90°  twice; 

B.  By  moving  the  elements  of  the  —  shell  perpendicularly 
to  the  common  line  of  centers  in  the  proportion  sin  6  (where 
^=cos~^  harmonic  displacement*  of  the  shell  element)  and 
moving  the  resulting  configuration,  a  directed  line,  one-half 
its  dimensions  toward  the  correlatively  reversed  position 
of  the  original  operand,  i.e.,  toward  the  position  of  the 
+  shell.  A  repetition  of  this  operation  would  produce  the 
+  shell. 

21.  Operation  A  is  excluded  by  reason  of  its  lack  of 
definiteness,  leaving  operation  B  as  the  operation  producing 
the  mean  reversed  state. 

Since  we  are  dealing  with  iodiographs  these  spatial 
operations  must  have  a  one-to-one  correspondence  with 
mathematical  operations  performed  upon  the  things  they 
symbolize. 

22.  Hence  a  mean  reversed  scalar  is  represented  in  all  its 
properties  by  a  directed  magnitude  in  space,  a  vector,  as  it 
is  called,  which  is  definitely  directed  as  soon  as  the  -|-  and 
—  shells  are  posited. 

23.  A  vector  is  any  magnitude  having  direction  of  exten- 
sion in  space,  a  directed  line,  plane,  etc.,  such  as  velocity, 
impulse,  force,  etc. 

Vectors  are  equal  when  they  possess  the  same  quantitative 


*  If  P  represents  an  element  of  the  shell, 
OA,  the  projection  of  the  radius  on  a  given 
diameter  is  its  harmonic  displacement, 

0  =  cos~^  OA. 


12  QUATERNIONS 

and  qualitative  properties,  viz.,  magnitude,  sense,  and 
direction  of  extension.  Direction  of  extension  is  that 
property  which  prevents  the  vectors  from  coinciding  (in 
whole  or  in  part)  when  brought  together. 

Parallel  vectors  of  the  same  length  are  equal.  Vectors 
can  be  made  coinitial  without  altering  their  properties. 

24.  It  is  customary  to  indicate  the  unit  vector  in  a  given 
direction  by  Greek  letters  a,  p,  y,  .  .  .  Generally  three  unit 
vectors  at  right  angles  to  each  other  are  assumed  as  reference 
units.     These  are  designated  by  i,  /,  k. 

25.  So  far  we  have  recognized  two  kinds  of  magnitudes, 
scalars  and  vectors,  and  six  operations,  addition,  subtrac- 
tion, reversion,  multiplication,  division,  and  mean  reversion. 

Applying  these  operations  to  scalars  we  find  that  they 
all  produce  scalars  again,  except  in  the  case  of  mean  rever- 
sion, and  that  produces  a  vector.  This  gave  us  the  second 
kind  of  magnitude,  to  which  we  will  now  proceed  to  apply 
the  six  fundamental  operations. 

26.  Reversal  is  merely  the  turning  of  the  vector  into  the 
opposite  direction,  as  the  word  implies.  The  result  is  some- 
times called  a  revector. 

27.  Addition  and  subtraction  of  vectors.  Subtraction  of 
vectors  is  merely  addition  with  the  minuend  reverted. 
Vectors  are  of  the  nature  of  strokes,  with  the  property  of 
absolute  direction  added.     The  laws  governing  the  addition 

of  strokes  evidently  hold 
here  also.  Thus,  vector  ad- 
dition is  commutative  and 
associative,  and  this  whether 
the  vectors  are  coplanar  or 
not.     Thus 

=  /-'' +«"+/?'',  etc., 


SPACE   IDIOGRAPHS  13 

where  a,  /?,  y  are  the  three  edges  of  a  parallelepiped.     The 
same  reasoning  would  apply  to  additional  vectors. 

28.  The  following  equations  are  self  evident : 

a  -\-a  +«  ...  to  m  terms  =ma, 
—  «  +  (  —  «)+...  to  m  terms=m(  — a)  =  —ma, 

29.  If  a,  /?,  ;-  be  three  coinitial  vectors,  then  any  fourth 
coinitial  vector,  ^,  can  be  expressed  as 

^  being  the  diagonal  of  the  parallelepiped  whose  edges  are 
xa,  yP,  and  zy. 

30.  If  a  is  a  unit  vector,  and  ma=A,  then  m  indicated 
generally  by  the  symbol  TA,  which  expresses  the  length  of 
the  vector  A,  is  called  the  tensor  {tendere,  to  stretch)  of  the 
vector  A.  a,  denoted  by  UA  is  called  the  unit  vector  of 
A.     Therefore 

A^TA'UA. 

Vectors  will  be  denoted  by  capital  Greek  letters  when  the 
tensor  and  unit  part  are  to  be  emphasized;  by  lower-case 
Greek  letters  when  the  question  of  length  is  not  important; 
and  by  the  corresponding  lower-case  English  and  Greek 
letters  when  speaking  of  the  tensor  and  unit  part  sepa- 
rately. 

Thus  the  same  vector  may  be  indicated  hy  A,  aa,  a. 

The  tensor  is  signless,  just  as  any  length,  the  height  of 
a  steeple,  for  instance,  is  signless;  or  the  height  of  a  man. 
A  man  cannot  be  —5  feet  tall. 


14  QUATERNIONS 


Exercises  in  Vector  Combinations 

31.  If  Za:+m/3=0,  then  1=0,  m=0,  for  in  no  other  way 
can  two  strokes  in  different  directions  cancel  each  other 
so  as  to  leave  the  pen  at  the  point  of  beginning,  unless 
a  =x^,  i.e.,  unless  a  is  parallel  to  /?. 

32.  If  Za+m/?=Zi^.+mi^, 
that  is,                     (Z  — Zi)a: +  (m  — mi)/3=0, 
then,  l=h,     m=Wi. 

33.  If  loL-^m^-{-nj=0,  and  Z,  m,  n  are  not  zero,  then  a, 
/?,  and  ;-  are  coplanar,  for  la  and  m/?  determine  a  plane 
which  contains  the  ends  of  iiy,  and  therefore  ny  itself. 

34.  If  Za+m/?+n;'=0,  and  Z,  m,  n  are  not  zero  but 
Z+m+7i=0,  then  (Z+m+n)a:=0,  and  subtracting  the  first 

equation,  we  get 

m{a—p)  -\-n{a  —  Y)  =0, 

whence  «— /?  and  a  —  y  are 
parallel  (§31).  But  a  —  y  con- 
nects the  ends  of  a  and  ;',  and 

a—p  the  ends  of  a  and  /?,  hence  i/  Z+m+n  =0,  «,  ^,  anc?  y 

terminate  in  the  same  line. 

35.  Conversely,  if  a,  /?,  and  y  terminate  collinearly  and 
la  +m/3  -\-ny  =0,  ^/ien  Z  +m  +n  =0. 

For  by  condition ,      a—p=x{a  —  y), 

or  {l  —  x)a—^-\-xy  =  0, 

in  which  1— a;— l+a;=0.  q.e.d. 


SPACE  IDIOGRAPHS  15 

36.  If    aa+h^-\-cy-\-  .  .  .  =dd,    then    evidently    a-\-h-\-c 
+  .  .  ,>d,oT        TA+TB  +  TC-\-  .  .  .  >TD, 

or,      Sum  of  the  tensors  >  tensor  of  the  sum,       21 T  >  Til , 
or     the  distance  a  man  travels  >  his  distance  from  home. 

Ti  the  vectors  are  parallel,     I1T  =  T^. 

37.  The  diagonals  of  a  parallelogram  mutually  bisect  each 
other. 

a=d+yy  =  j'-\-xd. 

.-.     §31, 

d=xdj         T=yTy 

Q.E.D. 

7-  and  d  being  parts  of  the  diagonals  to  the  point  of  inter- 
section, and  yy  and  xd  the  remaining  portions  respectively. 

38.  The  lines  joining  the  middle  points  of  the  opposite  sides 
of  any  quadrilateral,  whether  plane  or  gauche,  mutually  bisect 

each  other. 

" — ^  .       1       1 

One  bisector  is    a=—X-\-ii  +-^v. 

The  other  is         /?=|  +  i^+'^. 

Find  the  vectors  from  any 
assumed  point  to  the  middle  points  of  these  bisectors  and 
compare  the  results. 

Thus  the  vector  from  the  beginning  of  X  to  the  middle 
point  of  a  is 

„    ;  «  >^\n,^  ^v\  3.1  ^1 


16  QUATERNIONS 

The  vector  from  the  same  point  to  the  middle  point  of 


^-^+f+§ 


^+f4(f+'^+f) 


.3    ^1       1  k  +  fi  +  v     3.,  1     ,1 

.*.   '^i  =  s2,  and  the  middle  points  coincide.  q.e.d. 

39.  If  the  ends  of  two  parallel  vectors  be  connected  by 
straight  lines,  the  join  (connecting  line)  of  the  middle 
points  of  the  straight  lines  is  half  the  sum  or  difference  of 


-ma  met 


the  tensors  of  the  parallel  vectors:   i.e.,  the  median  line  of  a 
trapezoid  is  half  the  algebraic  sum  of  the  bases. 

8=-^±a—^,  taken  ^long  a ; 


ma±a 


P  T 

=  —  —  +  ma  H-^,         taken  along  ma. 


Whence  by  addition,     b  = 


2 


„     ^  ,     ma±a 

:.     §36,  d  =  ~-^,  Q.E.D. 


CHAPTER   IV 
MULTIPLICATION   OF   UNIT  VECTORS 

40.  Parallel  vectors.  Remembering  that  multiplication 
is.  the  performing  by  the  reader  on  the  multiplicand  of  an 
operation  which  is  symbolized  by  the  multiplier,  viz.,  the 
operation  which  produced  the  multiplier  from  unity,  we 
must  in  the  product  ii  *  ask  what  operation  is  the  first  i 
the  symbol  of.  The  answer  is,  of  course,  of  one  of  two  equal 
operations  whose  successive  applications  shall  produce 
reversal.  Now  i  can  be  reversed  by  the  repetition  of  each 
of  two  methods.  The  one  we  have  designated  as  operation 
A  (§  20).  The  other  we  have  designated  as  operation  B. 
Since  the  multiplier  is  exactly  the  same  as  the  multiplicand 
in  all  its  properties,  we  must  if  possible  use  exactly  the 
same  operation  not  only  in  kind,  but  also  in  detail,  that 
produced  the  multiplier,  that  is,  operation  B  (§  20). 

This  amounts  to  the  repetition  upon  the  multiplicand  i 
of  the  same  operation  which  produced  it  from  unity,  and 
of  course  results  in  —1,  see  §  20.     That  is, 

ii  =  —  \. 

Hence,  Multiplication  {'performance  of  an  operation  sym- 
bolized by  the  multiplier)  of  one  unit  vector  into  another 
parallel  to  it  produces  reversion. 

*  i=some  directed  v  —  l^some  directed  mean  reversed  scalar. 

17 


18 


QUATERNIONS 


41.  Perpendicular  vectors.  Since  the  multiplier  is  now 
perpendicular  to  the  multiplicand,  we  must  take  it  as  the 
symbol  of  an  operation  to  be  performed  on  the  multipli- 
cand, the  same  in  kind  but  as  far  removed  in  detail  from 
that  which  would  have  been  used  had  the  multiplier  been 
parallel  to  the  multiplicand  as  perpendicularity  is  removed 
from  parallelism. 

This  we  must  do  in  order  to  take  into  account  the  per- 
pendicularity of  direction  as  opposed  to  parallelism.     The 

operation  of  mean  reversion 
which  produced  the  multiplier 
was  operation  B  (§  20).  There- 
fore we  must  use  operation  A. 
Let  i  and  /  be  operator  and  ope- 
rand respectively.  The  only 
position  into  which  /  can  be  re- 
volved such  that  the  reversal 
of  the  signs  of  the  two  factors  will  give  the  same  result  is  kj 
one  of  perpendicularity  to  both  factors.     Thus 


ij=k       and 


■j  =  k, 


since  —i  bears  exactly  the  same  relation  to  —  /  that  i 
does  to  k,  and  must  therefore  have  the  same  effect.  Any 
other  position  than  k  for  the  product  of  ij  would  not  do 
this. 

Hence,  The  multiplication  {'performance  of  an  operation 
symbolized  by  the  multiplier)  of  one  unit  vector  into  another 
perpendicular  to  it  residts  in  the  turning  of  the  multiplicand 
through  a.  right  angle  in  (to)  a  plane  perpendicular  to  the  operator. 

'42.  Inclined  vectors.  Naturally  the  result  will  be  a  com- 
bination of  those  of  §§40,  41,  that  is  partly  scalar  and 
partly  vector,  or 

ap=—  cos  ^  +  £  sin  0, 


MULTIPLICATION  OF   UNIT  VECTORS 


19 


where  a  and  /?  are  the  two  unit  vectors  incKned  at  an 
angle  d,  and  e.  a  unit  vector  perpendicular  to  a  and  /?, 
since  this  formula  satisfies  both  the  limiting  cases  (§§  40, 
41). 

Hence,  The  multiplication  {performance  of  an  operation 
symbolized  by  the  operator)  of  one  unit  vector  into  another 
inclined  to  it  at  an  angle  0,  thus  produci7ig  the  mean  reversed 


state  induced  by  the  operator  symbol,  turns  the  operand  through 
a  right  angle  into  a  plane  perpendicular  to  the  multiplier, 
makes  its  length  sin  0  and  adds  a  sccdar,  —  cos  0. 
43.  We  can  get  the  same  result  as  follows: 
i^,  the  mean  reversed  state  of  /?,  must  be  as  to  direction 
some  vector  perpendicular  to  the  plane  of  i  and  ^,  since 
—  !•— /?  must  produce  the  same  result  as  i^.  Hence, 
tentatively, 

ip=sk, 

where  s  is  some  scalar.     Operating  again  with  i  to  see  if 
the  second  application  produces  reversal,  we  get 

i'i^=i-sk^S'ik=s-  —j, 

which  is  not  reversal,  but  which  would  be,  except  as  to 


20  QUATERNIONS 

length,  perhaps,  by  the  addition  of  —ci,  where  c  is  some 
scalar.     But  this  would  require 

ifi=—c+sk,         since  i-ip=—ic-\-i-sk 

=  —ic  —  sj. 
Now  if  —  ic  — s/=— /?,  then         c  =  cos^,     s=sin^^ 

and  we  have  as  before     i^  =  —  cos  d  -\-k  sin  d. 
44.  Exercises  in  unit  reference  vectors. 

ij  =  k,     but     ji  =  —k ;  jk  =i,     but     kj  =—i. 

Hence  the  factors  are  not  commutative, 

i-jk=i'i  =  —  l,  ij'k=k'k  =  —  l. 

Hence  i-jk=ij-kj  or  the  factors  are  associative. 
ki=j,  i-jk=ii=i^  =  —  l, 

ji  =  —k,  j-ki  ^jj  =  —  1, 

i-  —j  =—k,  ijk  =jki  =kij  =  —  1, 

jjlii  =jH^  =i,  k'ji=k-  —k  =  —k^  =  l. 

ii-k=kk=k^  =  —  l. 


CHAPTER  V 
QUATERNIONS 

45.  Having  ascertained  that  the  product  of  two  vectors 

a,  /?  is 

ap=—  cos  ^  -f-  £  sin  /9, 
we  can,  §  29,  express  £  sin  0  in  terms  of  i,  j,  k,  viz.: 
e^m  d=xi-\-yj+zk,     or     a^^—co&  6 -{-xi-\-yj-\-zky 

which,  being  composed  of  four  terms,  a  scalar  and  three 
vectors,  is  called  a  quaternion,  and  will  be  symbolized 
by  5. 

A  quaternion  is  evidently  composed  of  a  scalar  plus  a 
vector.  Later  (§  61)  we  shall  find  that,  conversely,  a  scalar 
plus  a  vector  is  a  quaternion. 

46.  The  plane  of  the  factors  (multiplier  and  multipli- 
cand) of  a  quaternion  is  called  the  plane  of  the  quaternion. 

The  plane  of  a  vector  is  the  plane  perpendicular  to  it. 

£  is  called  the  axis  of  a/?.  The  most  convenient  rnethod 
of  defining  it  seems  to  be:  The  unit  vector  toward  the 
north  pole  when  the  multiplicand  is  to  the  east  of  the 
multiplier,  the  equator  being  the  plane  of  the  quaternion; 
toward  the  south  pole  when  the  multiplicand  is  to  the 
west. 

47.  Meaning  of  — .     By  the  rule  for  division  we  must  first 

21 


22 


QUATERNIONS 


ascertain  what  must  be  done  to  a  to  produce  1.  Return- 
ing to  our  idoigraphic  shell;  to  convert  a  into  +1,  we 
must,  repeating  the  operation  which  produced  it,  apply 
operation  B  (§  20),  and  then  direct  it  one-half  its  dimen- 
*j    sions  towards  its  correlatively  reversed 

position.     Performing  these  operations 

on  the 


numerator 

we  get  by  application  of  B, 
and  then  by  directing  it, 

whence  — 


This  is  verified  by  the  fact  that  a-  —a  =  —a^  =  —  •  — 1  =1, 

hence  —  =  —a,  since  a— =  1,  being  a  functional*  operation 
a  a       ' 

followed  by  the  inverse  operation  and  therefore  resulting  in 

the  original  operand. 

Hence,  the  reciprocal  of  a  unit  vector  is  the  unit  vector 

reversed. 

48.  Meaning  off  4-,  ^,  etc.     In  a  similar  manner  -^=k, 


*  See  Appendix. 

/  1  1  .a/?lll 

t  —  means  ?  •  —  and  not  —  •  / .     Thus  we  can  write  —  —  =  a—/?—  =  a— 

i  i  I  P  r      ^  r      r 

=— .     But  we  cannot  write  —•  —=—,  for  ^— «—  =  /?(  — 7-) a (  —  .5)    does 

r  r  ^    r         r  ^ 

not  allow  the  5's  to  cancel  each  other,  the  vector  factors  not  being 
commutative,  §  44. 


QUATERNIONS 


23 


a 


and  ■3-=cos  ^— £  sin  ^,  since  this  satisfies  both  the  limiting 

cases  —  =  1  and  -^  =  +k,  0  being  the  angle  from  a  to  /?. 
%  1 


Examples  for  Practice 


—  k_.       ik  _ 


j  k     k' 


—  =k.       —i--~j  =  k 


-i 


^=i.    v^pk;^  =  -m'-    Yj  =  -^'-^  =  --J-j^l 


-7=1.  ~]==l. 

■k  y 

k      .  .i      . 

1  1 


j  k 


■I-  —1=1^ 


ikj  =  kji  ==  jik  =  —i'^  =  ~p  =  l. 


49.  Since       aa — =a-  —a=l= — aa  =  —a-a, 
aa  aa       *  ^ 

therefore  a  vector  is  commutative  with  its  reciprocal. 
60.  Since  ^=  — /?,  we  can  write 

1 


_/?=-^./?  =  l. 


^=aj  =  a(-^) 


Hence,  §  42, 

=  -cos  {n^6)  +£sin  {jz^O) 
=  cos  ^— £  sin  d, 
where,  as  before,  0  is  the  angle  from  a  to  ^, 


■H 


24  QUATERNIONS 

Similarly, 

pa  =  —cos  {  —  0)  +£  sin  {  —  0)  =  —cos  0  —  £  sin  ^, 

—a  =  —  cos  (tt  —  ^)  +  £  sin  {n  —  0)=  cos  6^  +  £  sin  ^. 


Hence  -^a 


r'^(?=4)' 


—^  =  —  cos  (tt  +  /9)  +  £  sin  (tt  +  ^)  =  cos  ^  —  £  sin  ^^ 

—  =  — cos  (jz  —  d)  +£  sin  (n—O)  =cos  6^  +£  sin  ^. 
a 

61.  Introducing  the  tensors  of  a  and  /?,  and  collecting  the 
results,  we  have, 

a/?=a6(  — cos  6  +  £  sin  ^),  ■^=^(cos  0  —  e  sin  ^), 

Ba=ab  (  —  cos  6^  — £  sin  ^),  —  =— (cos  6'  +  £  sin  ^). 

^  '  a     a 

52.  Distrihutivity  of  the  vector  multiplier.  Let  a,  /?,  ;-  be 
three  unit  vectors,  making  with  each  other  the  angles  (j),  0, 
2a,  as  shown;  (^  is  not  a  unit  vector. 

£  is  the  axis  of  a/?,  -q  oi  ay^  X^  oi  ad',  coplanar,  since  the 
the  planes  of  the  three  quaternions  have  the  common 
edge  a. 

P  +  r=^,  1^1  =2  cos  a. 

[|^l  means  length  of  d.     cos  a=^  diag.  of  the  parallelogram 

on  /?,  r-] 

a/?  =  —  cos  d  +  £  sin  0,  ay  =  —  cos  0  +  >?  sin  0- 

.*.     a^ +«;-=- cos  ^  — cos  ^+£  sin  ^  +  )^  sin  j>. 


QUATERNIONS 


25 


But  by  trigonometry,  since  the  angle  between  the  axes  is 
the  same  as  the  angle  between  the  planes  of  the  quaternions, 
and  since  a=a'  makes  the  sines  of  these  angles  propor- 
tional to  the  sines  of  the  adjacent  sides,  that  is,  sin  0,  sin  ^, 
therefore  £  sin  ^  +  >?  sin  0  will  lie  along  the  ^  axis  and 


(1) 


«/?  +  «;-  =  — cos  ^  — cos  (l)-\-xX^, 


where  x  is  some  unknown  tensor. 

(2)   But  «(/?  +  /-)  =a^  =2  cos  a(  — cos  0  +  1^  sin  0),  and  we 
now  have  to  show  that  this  agrees  with  a^-^ay. 


By  trigonometry, 


cos  0  = 


cos  (j)  =cos  i[}  cos  a  +sin  ^  sin  a  cos  0 

cos  ^  =  cos  ^  cos  a—  sin  ^  sin  a  cos  0 

cos  <j)  —  cos  0  cos  a  _  cos  (/»  cos  a  —  cos  8 
sin  (j)  sin  a  sin  9^  sin  a       ' 

cos<A+cos^  , 

,*.     — ?r- =cos  0. 

2  cos  a 


26  QUATERNIONS 

(3)  .*.     —2  cos  a  cos  ^  =  — cos  9^  — cos  ^. 

By  trigonometry  again, 
x^  =sin2  0  -\-sm^  cf)  +2  sin  0  sin  cj)  cos  A, 
cos  2a  =  cos  (j)  cos  ^  +sin  d>  sin  6  cos  A. 
.-.     z^=sm^  ^+sin2  0+2(cos  2a  — cos  cj)  cos  0) 

=  sin^  ^  +  sin^  ^  +  2  (cos^  a  —  sin^  a  —  cos  (f)  cos  6^) 
=  1  — cos^  ^4-1  — cos^  (j)+2  cos^  a  — 2+2  cos^  a 

—  2  cos  <^  cos  d 
=4  cos^  a  —  cos^  0  —  2  cos  ^  cos  <p  —  cos^  <^ 
=4  cos^  a—  (cos  ^S  +COS  /9)2. 
But 


,     ^ /-.      /cos  oS  +  cos  /^X 2    V  4cos2a— (cosQ^+cos^  (9)2 

sin^  =  \l-    — ^r^- = . 

^       ^        \      2  cos  a      /  ^  cos  a 

(4)  .'.     x=2  cos  a  sin  ^. 

.-.     «/?+«;-=— 2  cos  a  cos  0+2  cos  a  sin  ^- 1^      (1).(3),(4) 

=2  cos  a(  — cos  9^+ sin  (Jf-'Q 

=a(J^  +  r)^  Q.E.D.     (2) 

Hence  in   vector   multiplication,   the   multiplier  is   dis- 
tributive over  the  operand. 


QUATERNIONS 


27 


53.  li  al^ ^ab(- cos  0 -\-£  sin  0)  =q. 

ah  is  called  the  tensor  of  q  and  is  symbolized  by 
ab  =  Tq. 


—  ah  cos  0 


ah  sin  6  •  e 


ah  sin  d 


—  cos  ^4-£  sin  ^ 


cos  d 


sin  ^  •  e 


sin  d 


scalar  part  of  q  and  is  symbol- 
ized by  Sq. 

vector  part  of  q  and  is  symbol- 
ized by  Vq. 

tensor  of  the  vector  part  and  is 

symbolized  by  TVq. 

unit  part  and  is  symbolized  by 
Uq. 

unit  vector  of  the  vector  part 

and  is  symbolized  by  UVq. 

scalar  of  the  unit  part  and  is 

symbolized  by  SVq. 

vector  part  of  the  unit  part  and 
is  symbolized  by  VUq. 

tensor  of  the  vector  part  of  the 
unit  part  and  is  symbolized 
by  TVUq. 


Exercises 


54.  Show  that 
{TVq)^  =  -{Vq)^, 

{TqY  =  {Sq)^-{VqY, 


UVq  = 


TVq' 


TVUq^^. 


28  QUATERNIONS 

If  q=w+xi+yj-\-zk,  §  45,  show  that  Sq=w. 


Vq=xi+ijj-\-zk^  (1)      Tq=Vw^+x^+y^+z^, 

TVq  =Vx^  +7/2  +z2,  f/5  =  2-r^ 

V'M;^+a:^+?/^+z^ 


\/a;2 +2/2+^2' 

If  g  and  r  are  quaternions 


C/g 


(2)        T'qr  =  TqTr,        U-qr=UqUr,        U^ 

If  q  and  r  degenerate  to  a,  T-a^  =  —a^.         [a^ 

55.  By  §  54,  Eqs.  (1),  (2), 


T^r  =  V  W^  +X2  +  F2  +Z2, 
where 

r9=\/'w;i2+a;i2+?/i2+^i2^       Tr=\/^i^?T^2M^^^?+^2, 

.-.     Tf2+X2  +  F2+Z2 

=  (Wi^  -\-Xi^  +yi^  +^l2)  (W2^  +X2^  +2/2^  +2;22) , 

or   (Euler's  Theorem)    the  sum   of  four  squares  may   be 
resolved  into  two  factors,  each  of  which  is  the  sum  of  four 
squares. 
56.  Show  that 

VaP  =  -  Vpa,  ap-pa  =  {Sap)'^-(Vap)^ 

aP^Pa=2SaP,     .     (1)  ^{TaQ)^. 

(2)  Sx^x,      Sij=0,  Vij=k. 


QUATERNIONS 


29 


67.  Example.         a:+/?  =  7-. 

(a  ■VpY  =  f  =a^  +al3  ^^a  +/?2, 

or  §  56,  eq.  (1),      -a^ +2Sa^-h'^  =  -c^. 

If  this  is  a  right  triangle,  at  C,  then 
§  56,  eq.  (2),  ^a/?=0  and  a?-Vh'^=c^. 
If  not  a  right  triangle,  this  becomes 


^2^n2 


a?  -\-h'^  —  2ah  cos  c. 


(Law  of  Cosines.  Trig.) 


58.  Geometric  meaning  of  TVap  and  Sap. 

Va^=ab  sin  d-e, 
f^  f  TVap=ab  sin  0 


= parallelogram  on  a/?. 
Sa[^  =  —ah  cos  0 


=  —  (one  tensor -projection  of  the  other  upon  it). 

59.  Meaning  of  Sap;-.  Suppose  a,  p,  y  unit  vectors, 
^=  angle  between  a,  8,  and  0=  angle  between  ;-  and  plane 
of  a0. 

ap  =  —  cos  /9  +  £  sin  6, 

SaPr=S-{-cosd  ^-esmd)r 

=Sey  sin  d. 

But  S£y  =  —sin  0, 

.'.     Saj^j-  =  —  sin  ^  sin  6. 

If  a,  /?,  ;-  are  not  unit  vectors,  but  have  the  lengths 
a   h,  c,  respectively,  then 

Sa^f  =  —  ahc  sin  d  sin  <j) 

=  —volume  of  the  parallelopiped  on  a,  /9,  ;-,  as  edges. 


30  QUATERNIONS 

60.  If  Sal^-)'=0,  «,  /?,  ;-  are  coplanar,  and  vice  versa. 

61.  The  sum  of  a  scalar  and  a  vector  is  a  quaternion.     Let 

I/;  =  some  scalar;  A=aa  =some  vector. 

Then         w -\- afv  =\^ rn^ -{- a'-^ I     ^  - +<t    ^  | 

Ww^+a^       Vw^+a^/ 

=a  tensor  (cos  (j)+a  sin  0) 

=  a  quaternion.     (§  51.)  q.e.d. 

62.  Idiographic  proof.  We  can  always  construct  in  a 
plane  perpendicular  to  the  vector  a  right-angle  triangle, 
one  of  whose  legs  shall  equal  the  diameter  of  the  sphere 
which  is  the  idiograph  of  the  scalar,  and  the  other  leg 
the  tensor  of  the  given  vector.  If  we  denote  the  hypoth- 
enuse  of  this  triangle  by  a6,  then  the  leg  corresponding  to 
the  scalar  length  will  be  ah  cos  6,  and  the  other  leg  will  be 
ab  sin  6,  6  being  the  angle  adjacent  to  the  first  leg  con- 
structed. 

On  the  sides  including  6,  lay  off  respectively  the  distances 
a  and  6  as  vectors.  Then  the  product  of  these  two  vectors 
will  be  a  quaternion, 

—  a&  cos  ^+a&  sin  ^-e, 

which  will  be  the  given  sum,  and  at  the  same  time    the 
product  of  two  vectors. 

63.  We  are  now  able  to  distinguish  between  the  different 
branches  of  mathematics  of  discrete  magnitudes. 

First,  the  mathematics  of  numbers,  having  size  only. 
This  is  Arithmetic,  the  algebra  of  tensors.  The  operations 
are  four  only,  addition,  subtraction,  multiplication,  and 
division.     The  operands  are  tensors. 


QUATERNIONS  31 

Secondly,  the  mathematics  of  numbers  having  size  and 
sense,  using  in  addition  to  the  previous  operations,  reversion. 
Its  operands  are  scalars,  generally  symbolized  by  letters 
and  numerals.  This  is  Algebra,  in  its  broad  sense,  including 
Calculus  and  allied  subjects. 

The  usual  line  of  demarcation  between  arithmetic  and 
algebra,  the  use  or  non-use  of  literal  characters,  is  unphilo- 
sophical  and  erroneous  in  its  significance.  If  the  letters 
represent  tensors  (e.g.,  number  of  girls,  boys,  children), 
gf  +6  =c  is  arithmetic  purely,  and  4  — 5  =  —  1  is  pure  algebra. 

Thirdly,  the  mathematics  of  numbers  having  size  and 
sense,  and  which  adds  to  the  previous  operations  that  of 
mean  reversion.  This  is  the  algebra  of  Complex  Functions. 
Its  operands  are  scalars,  symboHzed  by  strokes  as  well  as 
by  letters  and  numerals. 

Fourthly,  that  branch  of  mathematics  which  deals  with 
numbers  having  size,  sense,  and  direction.  Its  operands 
are  scalars  and  vectors  and  combinations  of  these.  This  is 
the  subject  of  Quaternions. 


CHAPTER  VI 

KINDS   OF   QUATERNIONS 

64.  The  reciprocal  of  a  quaternion,  denoted  by  Rq,  is 
the  factor  into  which  the  quaternion  must  be  multiplied  in 
order  to  produce  unity. 

If  q=aab^, 

11  11*        1 

then  aabB-f-p, —  =aQyr  •—  =a—  =  l. 

^bp  aa        ^ p    a         a 

Hence  the  reciprocal  of  a  quaternion  is  found  by  taking 
the  product,  in  reverse  order,  of  the  reciprocals  of  the 
factors. 


6/3       "^aa 

1 

a/? 

1            1 

—         —                                                       —  rt/^o    H 

-e  sin  6. 

q      —  COS  0  +£  sm  u 

1    1 

—  COS 

^  +  £sin^.               .-.    -^7^--^. 
af:{     a  /? 

11111 
j^'a      a^^a'p' 

I{ap=q, 

^^4r^"- 

*  The  assumption  made  here  that  ^—•a  =  B-—a  is  justified  later, 

in  §90,  since  all  vectors  are  (§  97)  quaternions. 

32 


KINDS   OF  QUATERNIONS 

R(xl^=-(-cosO-£sm  d),       ^'-  =  1, 
^     ab                              ' '       ^   a       ' 

<^1- 

a   1      a     1       a   1  ^             ^     1         1. 

33 


Hence,  a  quaternion  and  its  reciprocal  are  commutative: 

65.  Opposite  quaternions  are  those  in  which  one  factor  is 
reversed. 

a^=ab{  — cos  d+s  sin  6) 

=ab(cos  {n-d)  +£  sin  {n-O)),  y^^ 

a{-p)  =ab(cos  d-ssin  d).  y  a     '* 

Hence  if  a/?=g,  a{  —  ^)=—q  '^ 

and  g  +  (-5)=0, 

or  the  sum  of  two  opposite  quaternions  is  zero. 

or  the  quotient  of  two  opposite  quaternions  is  —  1. 
Tq  =  T{-(i)=ab. 

Opposite  quaternions  have  a  common  plane,  equal  tensors, 
supplementary  angles  and  opposite  axes. 

66.  The  Conjugate  of  a  quaternion,  denoted  by  Kq,  has 
one  of  its  factors  turned  an  equal  angle  across  the  other 
factor,   the   order  of    the   factors  not  being   changed:    it 

reverses  the  angle  of  the  original  quater- 


^     nion. 


Kq=Kap=aP' 

=abi- cos  (-^)+£sin  (-0)) 
=ab(  — cos  d  —  e  sin  0). 


54  QUATERNIONS 

Kq=Sq-Vq.  Kx=x. 

q+Kq=2Sq.  ap=K^a. 

q  —  Kq=2Vq.  §55.      Kq=w-xi-ij]'-kz. 

Conjugate  quaternions  have  a  common  plane,  equal 
angles  between  the  factors,  equal  angles  (see  §  76),  equal 
tensors,  and  opposite  axes.     (§  46.) 

67.  Since         TKq  =  Tq     and     UKq=^^. 

.-.     Kq  =  TKq-UKq=^^ 
Whence  by  multiplication  and  division, 

qKq  =  {Tq)\  Tq^^^'^^'' 

Kq=Ka[^  =  Tq-^  =ah{-cos  O-ssm  6)  =pa. 

a     1 
.*.     Kap=^a       and       K-r=-ra. 

Evidently,    Kxq  =  xab {  —  cosO  —  £smd)=  xKq. 
Making  x  =  —  l,  we  have 

K{-q)  =  -Kq. 


68. 


1 

kI.kI.tI  '  . 

a         a        a  j-Ji 
J                         a 

1     -rh"- 

"f 

-^a   1     \x^r 

■•        9     Kq 

(X 

Confirm  this  by  diagram,  using  a/?  instead  of  -^. 


KINDS  OF  QUATERNIONS 


35 


69.  Collecting  the  q,  q^^,   —q,  and  Kq  into  a  table,  we 
have: 


T 

0. 

U 

a 

q 

—  (  —  cos  (b-\-  £  sin  c6) 

0 

Tq 

0 

Uq 

I 
a 

q~* 

—(  —  cos  96— £  sin  0) 
a 

1 

-0 

1 

—  a 

-q 

—(cos  9S— £sin  (56) 
0 

7^? 

7r+«9^ 

1 

Uq 

1 

r 

Kq 

—(  —  cos  ^  — £sin  ^) 
0 

Tq- 

-<i> 

1 

^1  indicates  the  angle  from  the  multiplier  to  the  multi- 
plicand, and  not  the  angle  between  a  and  /?. 

From  this  table  it  is  evident  that  if  ^=0,  KS=S,  i.e.. 
The  conjugate  of  a  scalar  is  the  scalar  itself. 

If  (f)=—,KV  =  —  V,  The  conjugate  of  a  vector  is  its  opposite. 

70.  RRq  =  (-)      =q,  The  reciprocal  of  the 

^  ^  reciprocal 

—  (  —  q)=q,  The  opposite  of  the 

opposite 
KKq  =g.  The  conjugate  of  the 

conjugate 

71.  Equality  of  quaternions.  If  a,  /?,;-,  <?  are  four 
coplanar  unit  vectors  with  the  same  angle  d  between  a  and 
/?,  and  ;-  and  d,  then 

a/?  =  —  cos  ^  +  £  sin  0,  yd  =  —  cosd  +  £  sin  ^, 

or  a/?  =  yd. 

Similarly,  -^=^=cos^  —  esin^. 


is  the  quater- 
nion itself. 


36  QUATERNIONS 

Hence,  Revolving  the  factors  of  a  quaternion  in  the  plane 
of  the  quaternion  does  not  alter  the  quaternion. 

Quaternions  having  the  same  S  and  V  parts  must  be  the 
same.  Diplanar  quaternions,  that  is,  quaternions  not 
having  their  factors  in  the  same  plane,  cannot  have  the  same 
V  part,  and  therefore  cannot  be  equal.  To  be  equal,  two 
quaternions  must  be  coplanar. 

72.  Two  quaternions  can  always  be  transformed  so  as  to 
have  the  same  vector  for  the  numerator  of  one  and  the 
denominator  of  the  other,  or  what  is  the  same  thing,  the 
multiplicand  of  one  the  multiplier  of  the  other. 

For  moving  the  quaternions  in  their  planes  until  the 
denominator  of  one  and  the  numerator  of  the  other  lie 


along  the  line  of  intersection  of  the  planes,  a  vector  along 
this  line  can  be  taken  as  the  numerator  of  one  quaternion 
quotient  and  the  denominator  of  the  other.     Thus 

a      u  r     XV      V 

p       V  -  o       p        p 

X 


CHAPTER   VII 
THE   QUATERNION    AS    A   MULTIPLIER 
73.  Into  the  reciprocal  of  its  multiplicand  factor. 


/5 


li=qp^a. 


Result:    The  quaternion  multiplier  turns  the  multiplicand 
through  the  angle  —0,'m.  the  plane 
of  the  quaternion. 

74.  Into  its  mulitplier  factor. 

cc 
qa  =-n(x  =«:  COS  d—ea  sin  0 

=a^Q:  =q:  cos  0  -\-ae  sin  0. 

[§51. 
Result:  Same  as  in  §  73. 

75.  Into  any  coplanar  factor ,  y. 

=  —xa{co^  0+£  sin  0)  +ya  = —xa  cos  d  —  x  sin  d'a£-\-ya. 
at  , 


{-xco8e-^v)cL 
Result:  Same  as  in  §  73. 


37 


38 


QUATERNIONS 


76.  Into  any  quaternion.  Let  the  quaternions  be  reduced, 
§  72,  to  a  common  numerator  and  denominator  respectively, 
viz.: 


Denote  the  arcs  which  de- 
termine the  planes  of  the  qua- 
ternions by  c,  a,  h  respec- 
tively, as  shown  in  the  figure. 
The  effect  of  the  quaternion 
whose  axis  is  ax-c  operating 
upon  the  quaternion  whose 
axis  is  ax -a  is  to  give  a  new 
quaternion  whose  axis  is  ax  •  h. 
This  can  be  analyzed  as  follows:  Suppose  we  move  the 
arc  a  horizontally  along  c,  being  careful  not  to  change  its 
inclination  to  c,  until  it  takes  the  position  a'.  Theaa;-a 
will  move  horizontally  around  ax  •  c  (conical  revolution)  until 
it  takes  the  position  ax -a'.  Then  as  we  turn  a'  down  to 
coincidence  with  h  around  a  as  an  axis,  ax  •  a'  will  move  up 
the  arc  passing  through  ax-c  (the  equator  of  a)  until  it 
takes  the  position  ax  •  b. 

That  is,  the  multiplication  of  ^  into  —  turns  the  axis  of  — 

(X 

horizontally,  i.e.,  parallel  to  the  plane  of  -^  (conical  revolu- 
tion) through  an  angle  equal  to  that  between  a  and  ,/?,  and 
raises  or  lowers  it  through  an  angle  which  depends  entirely 

upon  — .     Whatever  the  position  of  f  with  reference  to  5, 

the  horizontal  revolution  is  always  the  same,  dependent 
upon  a  and  8. 


THE  QUATERNION  AS  A  MULTIPLIER  39 

OL 

Hence,  the  effect  of  -^  as  a  multiplier  is  generally  to  revolve 

the  axis  of  the  multiplicand  conically  through  a  definite  angle 
around  the  axis  of  the  multiplier.  This  angle  {  —  0  here), 
the  angle  through  which  the  axis  of  the  operand  is  moved, 
is  called  the  angle  of  the  quaternion,  and  is  designated  by 
/  q,  or  in  these  notes  by  D. 

77.  Since  the  effect  of  a  quaternion,  g,  as  a  multiplier  is 
to  revolve  the  multiplicand  (axis)  through  the  angle  /.q 
around  the  ax-q,  the  arc  of  a  great  circle  which  measures 
Zg  on  the  equator  of  ax-q,  can  be  taken  as  an  indirect 
measure  of  Uq. 

Just  as  a  quaternion  can  be  revolved  in  its  plane  without 
alteration  of  value,  so  the  arc  representing  it  can  be  moved 
in  its  great  circle  without  alteration  of  value. 

Arcs  in  different  great  circles  cannot  be  equal.  If 
they  are  semicircles,  any  semicircle  will  have  the  same 
effect. 

78.  Representing  the  quaternions  -r,  — ,  — ,  by  their  arcs 

(arc  strokes)  c,  a,  b  respectively,  we  have,  considering  the 
arcs  as  arc  strokes  on  the  sphere  surface  similar  to  the 
plane  strokes  of  §  12, 

a  8     a  ,        T 

77'— =—       or       a-{-c=o, 

P  r    r 

and  we  have  reduced  the  multiplication  of 
quaternions  to  the  addition  of  arc  strokes. 
In  the  same  way 


a  r    r 


6  +  (-c)=a. 


40 


QUATERNIONS 


1^^" 


79.  Notice  carefully  that  the  arc  strokes  corresponding 
to  the  quaternion  factors  read  from  the  left,  are  added  up 
from  the  right. 

80.  Evidently  from  the  figure  qq'  is  quite  a  different  arc 
from  q^q,  and  hence,  The  addition  of  arc  strokes  is  not  commu- 
tative, unless  they  are  coplanar;  and  correspondingly,  the 
7nultiplication  of  quaternions  is  not  commutative,  unless  the 
quaternions  are  coplanar. 

Rule  for  multiplication  by  arc  strokes. 
To  the  arc  stroke  of  the  multiplicand 
add  the  arc  stroke  of  the  multiplier. 

Rule   for    division    by    arc    strokes. 
Reverse  the    arc    stroke   of  the  divisor 
and    add    the    arc    stroke   of    the  divi- 
dend. 
Show  that 

qKq^Kq-q.  q{-q)=-q.q. 

11  -1-1 

q-=-q=qq  ^  =q  \ 

81.  From  this  we  see  that  the  product  of  a  quaternion,  is  a 
quaternion,  the  sum  of  two  arc  strokes  being  an  arc  stroke. 

Take  two  quaternions  in  the  form  of  a  scalar  plus  a 
vector,  and  show  that  their  product  is  a  scalar  plus  a  vector, 
and  therefore,  §  61,  a  quaternion. 

82.  By  multiplying  the  various  forms  of  a  quaternion, 

a   B  . 

ap,  /?«,  ■^,  — ,  etc.,  into  some  vector,  whatever  will  give  the 

result  most  readily,  we  find  the  following  angles  for  the 
quaternions,  d  being  the  angle  from  a  to  /?,  Oi  the  angle 
from  the  multiplier  to  the  multiplicand,  and  D  the  angle  of 
the  quaternion. 


THE  QUATERNION  AS   A  MULTIPLIER 


41 


2 

D 

e. 

q 

D 

0^ 

a/? 

Tz-e 

d 

1 

d 

TZ-d 

/?« 

TZ+d 

-d 

a 

-0 

K  +  O 

I 
a 

-d 

TZ-\-d 

-«/? 

-e 

-TZ  +  d 

d 

iz-d 

Inspection  of  the  table  shows  that:  The  angle,  D,  of  a 
quaternion  is  the  supplement  of  the  angle  between  the 
factors  measured  from  the  multiplier  to  the  multiplicand. 

83.  Representing  the  angle  measured  from  a  to  ^  by  ^, 

and  the  angle  of  the  quaternion  by  D,  the  angle  of  a/?  by 

ex. 
(Jf,  the  angle  of  -^  by  (f),  and  abbreviating  cos  D+£  sin  D  by 

C£S  D,  we  get  the  following  table: 


a/? 

CcS  (p 

C£S  {71+ (l)) 

C£S  (tz-O) 

C£SD 

fia 

C£S  (271-^) 

C£S(7Z-c})) 

C£S  (tZ  +  O) 

CSS  D 

a 

1 

C£S  (7r+  (/f) 

CSS  <p 

ces{-0) 

C£SD 

I 
a 

CSS  (tz—  (/>) 

CSS  (27r-f/.) 

cesO 

C£SD 

1 

r 

CSS  (tz—  (p) 

C£S(-0) 

cesO 

cesD 

a 

C£S  {tZ-\-  4>) 

C£S  </> 

ces(-O) 

cesD 

42 


QUATERNIONS 


84.  Since  a  vector  is  the  special  case  of  a  quaternion,  one 
whose  angle  is  — ,  we  can  represent  a  vector  by  an  arc  stroke 


of  —  on  its  equator.     As  an  example 


take  the  case  -^y. 


a 

arc  -A=c, 


arc  y=a, 


:/'=a+c  =6  =arc  <?, 


and  y  has  been  revolved  through  Z  ^. 

85.  Collecting  the  results  of  these  articles  we  have  the 
table  on  the  following  page,  where 


di=7t-D,         D=lq,         il^-^lap, 


*=z|  =  /l., 


d=  L.  —  =  /_ 77a.       d\  =angle  from  first  factor  to  the  second. 
a         [i 

0  =  angle  from  a  to  /?. 

86.  Since  quaternions  are  multiplied  by  adding  their  arc 
strokes,  the  square,  q^,  of  a  quaternion  will  have  for  its 
representative  arc  stroke  twice  the  stroke  of  q.  Similarly, 
one-half  the  arc  stroke  will  be  representative  of  qh.  But 
since  there  are  two  arcs  D  and  —  {2n  —  D),q^  will  have  two 

representative  strokes,  ^D  and  —71 -{-—-,  either  of  which  if 

A 

doubled  will  give  the  stroke  of  q.  Hence,  Rule  for  extraction 
of  square  roots  of  unit  quaternions:  Halve  the  angles  of  the 
quaternion. 


THE  QUATERNION  AS  A  MULTIPLIER 


43 


w 

«> 

«o 

Ui 

(0 

<o 

01 

«C> 

<as 

• 

t> 

!? 

<ji 

-§ 

1 
I 

1 

1 

1 

1 

CO 

<3i 

■^s 

02 

m 

«15 

<^ 

<55 

<IS 

CO 

t? 

o 

s 

o 

o 

§ 

O 

o 

1 

1 

e  l-o 

^1  c 

el^ 

-o  1  e 

^ 

^1  « 

-1^ 

^1  « 

«1<^ 

1    ^ 

-e- 

-e- 

-e- 

-e- 

+ 

1 

-0- 

1 

-0- 

1 

-0- 

w 

K 

ti 

1 

-^ 

-^ 

-^ 

-^ 

-^ 

-^ 

NJ 

-S- 

1 

+ 

1 

+ 

1 

+ 

K 

k 

K 

ti 

ti 

Qi 

<5S 

■5^ 

<& 

<3i 

1 

+ 

1 

<5i 

1 

CO 

1 

k  |(N 

t: 

k 

^ 

^ 

« 

-o 

e 

c  l-o 

ro  1  e 

^1  e 

e    IrO 

-§ 

rH 

■3  "5 

c& 

<3i 

<3S 

<^ 

<C> 

1 

+ 

1 

<3i 

1 

<& 

1 

H  l(N 

Q  s 

ti 

t; 

1 

1 

/— N 

<& 

<3S 

1 

<5i 

C3 

■55 

<^ 

<^ 

<3i> 

.5 

'm 

c 

fl 

c 

^ 

•  S 

•1-1 

*> 

*M 

'm 

m 

n 

*m 

K?^ 

4- 

1 

Ui 

U) 

u> 

x> 

«« 

<u 

!::) 

•5^ 

<I5 

+ 

+ 

1 

+ 

1 

tn 

22 

^ — s 

<& 

■^s 

<I5 

<5i 

O 

O 

<IS 

03 

m 

o 

t» 

I 

O 

O 

Q 

o 

1 

1 

•>— / 

o 

« 

« 

« 

02 

o 

^ 

■Si 

1 

CO 

<C. 

qii 

<^ 

<C) 

<3i 

+ 

1 

+ 

1 

+ 

klc^ 

1 

ti 

K 

1=; 

ti 

tJ 

<51 

4 

«|Q3. 

Q3.I    « 

^1  « 

1 

<» 

44  QUATERNIONS 

87.  Since  the  angle  of  the  quaternion  is  the  supplement 
of  the  angles  measured  from  the  multiplier  to  the  multi- 
plicand, that  is,  7t  —  D=6i,  a  better  working  rule  would  be: 

For  square  root.  Revert  the  multiplicand  and  halve  the 
angles  between  this  and  the  multiplier,  for  the  factors  of  qi. 

Similarly,  a  working  rule  for  squaring  quaternions: 
Double  the  angle  between  the  factors  and  revert  the  multipli^ 
cand  in  its  new  position  for  the  factors  of  q^. 

In  ^2  =py,  or  -  =a|^,  ^  bisects  the  angle  between  /?  and  ;-. 

88.  Using  the  arcs  of  quaternions,  show  that 

(1)  arc  (72  =  arc  g+arc  g=2  arc  9±2n;r. 

(2)  arc  g^  =  }  (arc  q  ±  2nn) . 

(3)  arc  g^=arc  y+arc  q-{-  .  .  .  =n  arc  q±2n7t. 

i     1 

(4)  arc  g^  = — (arc  q  ±  2n7i) .  (5)     (pg)  ^  ^^  p^^. 

89.  When  the  quaternion  is  a  scalar,  its  arc  is  some 
multiple  of  n.  All  coinitial  arcs  of  the  same  length,  n,  for 
instance,  will  represent  will  represent  the  same  scalar,  —  1, 
however  the  arcs  are  situated;  that  is,  whatever  the 
direction  of  their  axes.  In  halving  these  arcs  to  extract 
square  roots  we  will  get  an  infinite  number  of  answers, 
one  for  each  arc,  each  corresponding  to  its  own  axis.  Or 
in  other  words,  there  are  an  infinite  number  of  vectors  the 
square  of  which  is  a  scalar. 

Similar  reasoning  shows  that  there  are  an  infinite  number 
of  quaternion  nth  roots  of  a  scalar.  On  this  account  the 
roots  of  a  scalar  are  limited  to  scalars. 

From  Eq.  (2)  of  §  88  we  get  (pg)^  =  cos  —  +£  sin  — ; 

repetition  of  which  gives  (/>g)°  =  cos  0  =  1, 

thus  illustrating  for  the  quaternion  number  the  familiar 

fact  of  algebra. 


CHAPTER   VIII 
PRODUCTS   OF    QUATERNIONS 

90.  Having  three  quaternions,  let  the  first  two  be  reduced, 
§  72,  to  a  common  denominator  and  numerator  respect- 
ively, viz.,  q=-n,  ^=— ;   and  the  latter  two  also,  viz.,  r=— , 

s  =— ,  where,  evidently,  since  they  represent  the  same  quater- 
nion, 

r    ^' 

The  product  of  the  three  quaternions  is 

■qr8. 


At    ^}p 


Associating  the  first  two  factors,  we  have, 
a  Q   V      a  V  11 

?r  p    r  p       r   p 

Associating  the  second  two,  we  have 

a  I  ti   v\      a    a  1      1 

/?\y   p)      P  p  r  P 

But  if  ^=^  then  ^=A 

45 


46  QUATERNIONS 

Multiplying  this  by  -  and  into  //,  we  have 
1        1 

Whence  =-t,-—       or       qr-s^q-rs, 

r  P     ?  P 

or  quaternions  are  associative  in  multiplication. 
91.  Since,  §  67, 

Kq  =  -^  =  Tq{-cos  O-esin  0)  =Tq(-cos(-0)  +£sin(-^)), 

the  arc  representing  Kq  will  be  the  reverse  of  that  repre- 
senting q,  or  arc  Kq  =  —  arc  q. 


ar 


Inspection  of  the  figures  shows  that 

—  arc  gr  =  +arc  KrKq. 
Hence 

K(qr)  =KrKq,  or 

The   conjugate   of  the  product   of  two   quaternions  is   the 
'product  of  their  conjugates  in  a  reverse  order. 

K{aP)  =  -p-  -a=da.       Conf.  §  67. 

92.  Coplanar  quaternions.     Using  arc  strokes  for  q  and 
r  show  that 

qr=rq.  Conf.  §  80. 


PRODUCTS   OF   QUATERNIONS  47 

Therefore,  since  a  quaternion  and  its  reciprocal,  opposite, 
conjugate,  and  any  power  are  coplanar, 

qKq=Kq-q.         q~'^  •  —q  =  —q-q~'^.          Kq-q'^^q^Kq. 

93.  Using    q=Sq  +  Vq,    r=Sr  +  Vr,    show    that    qr    is    a 
quaternion,  the  sum  cf  a  scalar  and  a  vector. 
Show  that 

Sqr  =SqSr  +8-  VqVr.  V  •  VqVr  =  -V-  VrVq. 

Vqr=SqVr-\-SrVq-\-V'VqVr.  qr^^rq.       unless  coplanar. 

Srq  =SrSq  +S  ■  VrVq.  T-qr^  TqTr. 

Vrq^Sr^q  +SqSr  +  V  •  VrVq.  V-qr  =  VqVr. 

Srq  =  Sqr.  S-qrj^  SqSr^ 

unless  S'VqVr^O,  i.e.,  unless  the  planes  of  the  two  qua- 
ternions are  perpendicular. 

Vqr  7^  Vrq, 

unless  V •  VqVr^O,  i.e.,  unless  the  quaternions  are  coplanar. 
Since 

(p  +^)  ir +s)  =V^ +?^ +ps  +5^  =P^ +ps  +9^ +?^  =etc., 

therefore  the  distributive  and  associative  law  applies  to 
quaternions,  verified  by  resolving  the  quaternions  into  their 
vector  factors  and  applying  the  results  of  §  52. 


CHAPTER  IX 

VERSORS 

94.  i[/a/?  =  — cos  ^+£  sin  ^=cos  D+£  sin  D,  turns  any 
quaternion  to  which  it  is  applied  as  a  multiplier,  horizontally 
through  the  angle  D. 

li  D=—  then  Ua^=£^  turns  the  multiplicand  quaternion 
through  the  angle  — .     Applied  again  it  turns  it  through 

Zi 

another  right  angle,  or  in  all  through  the  angle  n.  That  is 
£  turns  the  multiplicand  through  one  rt.  Z ,  £^  turns  it 
through  two  rt.  Z  's,  and  so  on.*  Hence,  by  the  law  of 
mathematical  continuity,  e^  should  be  the  symbol  of  turning 

2 

through  J  rt.  Z ,  etc.  If  ^  is  expressed  in  radians,  z  "^  turns 
through  the  angle  0. 

If  £  is  perpendicular  to  its  operand,  we  recognize  the 
familiar  equations, 

ij =kj     i^j  =  —  /,     i^j  =  —k,  etc. 

According  to  this  notation  then, 

2 

UaS  =  cos  7)  +  £  sin  7)  =  £  'f    . 


♦Remember  that  e^  does  not  mean  here  e  multiplied  by  itself,  but 
e  applied  twice  to  some  operand.     See  Appendix. 

48 


VERSORS  49 


.2-D 


96.  Since  e  and  —e  are  opposed,  £    ""    ,  which  turns  clock- 
wise from  the  upper  side  of  the  plane,  must  have  the  same 

effect  as  ( —  £)  '^    ,  which   turns  counterclockwise  from  the 
lower  side  of  the  plane.     Hence 

r"^''  =  (-£)^'',        or         e-^  =  (-£)^. 

If  D  =-Jr,  then  £~^  =  —  £,  or  to  put  it  in  the  more  familiar 

form,  i~^  =  —i. 

96.  Again,  §  85, 

8  -e 

U-  =  c,os  d-\-£  sin  d^e""    , 
a  ' 


and 


C/|=cos  (-0)  +£  sin  (-0)  =£ 


therefore,             £~^=-t. 

From  these      £- ^=cos  (  — 6^) +£sin  (- 

-d)=(-s)^, 

—  (s^)  =  —  cos  d  —  £  sin  d. 

Hence  in  general. 

{-e)'  =  e-<>^-U'), 

unless  ^  =  1,  in  which  case,  as  before,  e  ^  =  — e. 

97.  The  unit  portion  of  a  quaternion  (considered  as  the 
symbol  of  an  operation  to  be  performed  by  the  reader) 
results  in  the  turning  of  the  operand,  and  is  therefore  aptly 
termed  a  versor. 


60  QUATERNIONS 

It  must  not  be  forgotten  that  all  vectors  (considered  as 
symbols  of  an  operation  to  be  performed  by  the  reader) 
are  versors,  as  indeed  all  vectors  are  special  cases  of  qua- 
ternions. The  evanescence  of  the  scalar  part  does  not 
affect  its  versorial  character. 

i,  j,  and  k  are  called  quadrantal  versors. 

98.  The  method  of  expressing  the  unit  part  of  a  quater- 
nion as  the  power  of  a  vector  gives  a  convenient  method  of 
indicating  multiplication  and  division  of  coplanar  quater- 
nions. 


Uq- 

=  COS(j)  +i 

■sm(j)=£''   .         Uq'  = 

=  COS 

(p+£  sin  (/f  = 

1^ 

£""    . 

•*• 

UqUq'  = 

=  (cos  ^+£  sin  (j))(cos 

'P+ 

£  sm  ([>)  =£^ 

=  cos  {(f) +(p)  +£  sin  (9^+^) 

2 

Similarly, 

e<f>.e-~'^=^  =  £<l>- 

.4, 

t 

id 

(£«^)m=^m^^ 

Hence,  The  algebraic  law  of  indices  holds  good  for  versors. 
Hence,  The  angle  of  the  product  of  two  coplanar  quaternions 
is  the  sum  of  the  angles  of  the  two  factors. 
99.  According  to  this  notation, 

a^=4:{- cos  30°  +£  sin  30°) 

=4(cos  150° +s  sin  150°)  =4A 

a/?  =2  (cos  60°  -  £  sin  60°) 

=2(cos  (-60°)  +£  sin  (-60°))  =2£-* 


VERSORS 


51 


4i 

Show  that  (f  =  \  Tqicos  9S  +£  sin  0)  {2  =  Tq)^e  ^ 
=  (Tq)^ (cos  2(l>+e  sin  2cl>). 
q"'  =  i  Tq) "  (cos  n<j)-\-s.  sin  n<f)) . 
Hence  any  quaternion  can  be  written,  where  p  is  some 


vector, 


24> 


p'=p-. 

100.  In  the  diagram  to  §  57, 

2A                                  2B 

2C 

2(A  +  B  +  C) 

,\     £        "^         =  —  1       or 

A+B+C  =7r. 

101.  What  will  convert  rq  into  qrf  Assume  for  the  moment 
that  the  tensors  are  unity.  Let  the  strokes  of  q  and  r  be 
as  shown.  Then  the  strokes  of  qr 
and  rq  will  take  the  positions  shown. 

From  the  table  of  §  83,  we  find 


9  = 


qr 


r^-^n 


and  the  problem  becomes:    To  con- 

vert  —.  into  — .     Now 
a  y 


a     a  ^ 

a   y'    a' 

_a  r'    /? 
^  a'   a 

§71. 
§71. 


Or  substituting  for  these  their  equivalents,  §  69, 
qr=q{rq)q-^. 


52 


QUATERNIONS 


102.  Meaning  of  q(r)q~'^.  Indicate  q  and  r  by  their  arc 
strokes,  as  shown  in  the  diagram.  The  other  strokes  will 
have  the  values  indicated. 

The  two  triangles  have  two  sides  and  the  included  angle 

respectively  equal,  and,  as 
shown  in  the  figure,  by  the 
angle  6,  qrq~^  is  r  revolved 
in  the  plane  of  q  through 
2  /_q.  This  amounts  to  re- 
volving the  axis  of  r  conic- 
ally  around  the  axis  of  q 
through  2  Z  g. 

Hence  qrq~'^  differs  from 

r  only  in  being  rotated  through   a   certain   angle.      Hence 

q{  )q~^   may  be   aptly   called   a  rotator,   since  it   rotates 

any  quaternion  inserted  in  the  parenthesis. 

This  is  a  special  case  of  a  more  general  function,  called  a 

nonion,  which  we  shall  meet  with  farther  on. 

103.  If  r  is  some  multiple  of  q,  say  q^q,  then 

q{q'q)q-^=qq'qq-^=qq\ 

Hence  qq'  is  q'q  rotated  through  the  angle  2  Z  ?  in  the 
plane  of  q. 

Compare  this  with  the  figure  of  §  80,  and  see  how  they 
agree. 

Similarly,  q~hq  rotates  r  negatively  through  2  /_q. 

Since,  Tq-T-^l,     therefore,     T-qrq-^ 


Tr, 


so  that  in  the  discussion  above  only  the  unit  or  versor  parts 
needed  to  be  considered. 

104.  qk-'=^', 

revolves  the  plane  of  /?,  or  /?,  conically  around  the  axis  of 
q  through  2  Zq- 


VERSORS  53 

106.  apa-^=p\ 

revolves  /?  conically  around  a  through  the  angle  2;r,  that  is, 
turns  /?  in  the  plane  of  a/?  to  a  corresponding  position  on 
the  other  side  of  a. 

106.  Exercises.     If  oiy=q,  show  that  aqa~^  ^Kq. 

If  a ,  ^,  7-  are  coplanar,  show  that  o:  •  /?;-  •  a:  "^  ==  a  '^^ya  =  K^y. 

rqBq~h~^=rq-B'{rq)~^. 


CHAPTER  X 


INTERPRETATION   OF   VECTOR  EQUATIONS 


107.  a:/?;'=a/?2/?-Y 


a,    /?,    ;-    coplanar,    i.e., 


Sa^-jr=0.     U -  —-X  acting  on  /?  turns  it  through  an  angle  0 

into  —a.     Acting  on  ;-  it  turns  it  through  the  same  angle 

into  d.     li  a,  p,  Y  represent  in  direction  the  successive  sides 

a. 
of  a  polygon  (which  is  always  possible)  then  U-  —-^  must 

turn  ;-  into  a  direction  coinciding  with  the  fourth  side  of 
a  polygon  inscribed  in  a  circle  drawn 
through  the  intersection  of  a,  /?,  and 
y.  The  diagram  shows  this.  Hence 
d  is  in  direction  the  fourth  side  of 
the  inscribed  polygon  of  which  a,  /?, 
and  ;-  are  the  other  three. 

If  the  circle  passes  through  three 

intersections,  that  circumscribes  a  /?,  ^  as  a  triangle,  then 

d  becomes  tangent  to  this  circle. 
108.  Locus  of  ^  in  [^=a$. 


^=a-^l^=Sa-'^^  +  Va-^l^. 


Sa-'^^=0,     e  =  Fa-i/?, 


and  ^  is  a  constant  vector  perpendicular  to  a  and  to  ^3,  and 
therefore  locates  a  point. 

54 


INTERPRETATION   OF  VECTOR  EQUATIONS  55 

109.  Locus  of  e  in  7ae=/?. 

/?±a.        a^  =Sa^  +  Va$  =x  +/?. 

.*.     ^=a~^x-\-a~^^=za-\-j'.         [since  a _L/?, 

and  the  locus  of  l^  is  a  vector  through  the  point  determined 
by  ;'(_L«,  _L/?),  and  parallel  to  a. 

Notice  the  difference  in  the  reading  and  interpretation 
of  the  equations  of  this  and  the  previous  section.  In  one 
/?  is  the  whole  of  a^,  in  the  other  only  the  vector  part. 

110.  Locus  of  e  in  Sa$=0,  Sal^=0,  Sj^^^-c. 

Sa^=0  limits  $  to  the  plane  of  a. 
Sal^=0  puts  /?  in  the  same  plane. 
S^^  =  —c       or       6a;cos^=c, 

where  b  and  x  are  the  tensors  of  /?,  $  respectively,  makes  the 

c 
projection  of  I"  on  /?  a  constant,  viz.,  x  cos  0=-r.    Therefore 

the  locus  of  ^  is  a  line  J_  to  /?  in  the  plane  of  a  and  through 

c 
the  point  of  ^  distant  from  the  origin  -r- 

111.  From  the  table  of  §  85,  we  find 

a/?  =  cos  ^  +£  sin  (f).  (a  +^) (a  +/?)  =  (a  +/?)2 

/?«  =cos  0  — £  sin  ^.  =a^+a^+^a+^^. 

a^-^a  =2Vap  =  -2F/?a. 

112.  By  §§  53,  54,  the  last  equation  of  §  111  becomes  the 
well-known  formula  for  triangle, 

c2=a2  4-62-2a6cos^. 


56 


QUATERNIONS 


113.  As  in  §  111,  (a:-/?)2=a2_2^Q;^+^2^ 


=a2_2Fa/?-/?2. 

a^.pa  =  {Sap  +  7a:/?)  (Sa/S -  7a:/?) 
=  (>S'a/?)2-(7a/?)2 
=a262  cos2  D  -  £2  sin2  D  •  a262 
=a262(cos2  7)+sin2  2)) 
=  {Tap)^. 

ap'^a^aP'^a        by  the  associative  law. 
==a:2/?2         since  /?2  is  a  scalar. 
7^(a/?)2.      Conf.  §  88,  (5). 
ap-ap  =  {aP)^  a:^-a:/?=cos2  Z)  +£  2sin  D  cos  i)-sin2  Z) 

=  (*S  +  7)2.  =cos  2/)  +£  sin  22).      Conf.  §  98. 

114.  Applications.     //  the  diagonals  of  a  parallelegram  are 
perpendicular  to  each  other,  the  parallelogram  is  a  rhombus. 

By  hypoth., 

'^"^^          ^(a: +/?)(/?- a)  =0  §56. 

=,Q^-a^.  §  113. 

.-.      {TpY  =  (TaY.  Q.E.D. 

115.  The  joins  *  of  the  mid-points  of  the  sides  of  a  rhombus 
are  at  right  angles. 

=  }(27a:/?+/?2-o:2).  §  113. 

.*.     Sr^=\{p^-a^)  =0.        [since  Ta=Tp. 

•  *.       T-L^-  Q.E.D. 

*  Join  =  the  line  joining. 


INTERPRETATION  OF  VECTOR  EQUATIONS  57 

116.  In  any  plane  triangle  to  find  a  side  in  terms  of  the 
other  two  sides  and  the  opposite  angles. 

Multiplying  by  (or  into)  a,  (or  any 
vector,  in  order  to  get  a  quaternion) 

Taking  the  scalar  parts,  we  have 
-a^+Sal^=Sar.  -a^-ab  cos  (180°- C)  =-ac  cos  B. 

.'.     a=b  cosC +  c  cos  B.         (Conf.  Trigonom.)  q.e.i. 

117.  Had  we  taken  the  vector  parts  in  §  116,  we  would 
have  found 

Va^  =  Vay.  ab  sin  C •£=ac  sin  B- e, 

or  6  sin  C  =c  sin  B.  (Conf.  Trigonom.) 

118.  In  §  116,  had  we  divided  by  some  vector  in  order 
to  get  a  quaternion,  say  y,  then  we  would  have  had  the 
same  result,  c=a  cos  B-\-b  cos  A. 

119.  Had  we  divided  by  a  or  /?,  say  a,  then  if  the  triangle 

were  right-angled  at  C,  since  *S— =0,  we  would  get 

l=S—=—  cos  B,        or        cosB=—. 
a     a  c 

120.  Had  we  taken  the  vector  parts  in  §  118,  we  should 
have  had  a  sin  B^b  sin  A.    (Law  of  Sines,  Trig.) 

Had  we  taken  vector  parts  in  §  119,  then  since  sin  C  =  l, 

sin  5=-^. 
c 

121.  V  T  is  representable  by  a  prolate  spheroidal  shell 
with  its  major  axis  i  and  its  two  minor  axes  ^j  and  ^k. 


58  QUATERNIONS 

Repetitions  of  this  process  tend  toward  the  unit  shell  and 
i°  =  l. 

In  the  same  way  \/^=\/a\^T  is  represent  able  by  a 
prolate  shell  whose  major  axis  is  i  and  whose  other  axes 
a-re  ij,  ik,  leading  similarly  to  (a/?)°  =  l. 

From  §  113,  since, 


\/(a/?)2  =  \/cos  2D+£  sin  2D  =cos  D+e  sin  D. 
'.'.     (a/?)°=cosO°  =  l, 
the  same  as  above,  and  agreeing  with  §  89. 
122.  Formulae  for  reference  and  practice. 

0  =  angle  from  a  to  ^. 

i)=  angle  of  the  quaternion.  [§  76. 

p,  q,  r  .  .  .  =  quaternions. 

1.  q  =  Tq(- cos  d  +  e  sin  0)  =a^. 

2.  =Tq{cosD  +  esinD). 

2 

3.  =  Tq-Uq^Tqe-^.  4:.  q=Sq  +  Vq. 

2 

5.  Kq=Sq-Vq  =  Tq(cos  D-e  sin  D)  =Tq£~^^. 

6.  Sq  =  Tq  cos  D  =  Tq(- cos  0). 

7.  TVq  =  Tq  sin  d ^Tq  sin  D. 

8.  Vq  -  TVq-  UVq  =  Tq  sin  D  •  s. 

9.  £^+^'.  10.  Sq=i(q+Kq). 

11.  Vq=i(q-Kq). 

12.  {Tq)^=qKq=Kq'q  =  {SqY-{VqY. 

13.  (rg)2  =  (^g)2  +  (r7g)2.  20.  ra2  =  _a2. 


»«--<^v 

21.  5q:=0. 

22.  7a=a:. 

15.  KKq=q, 

23.  KaP=^a. 

lG.Kx=^x. 

24.  Sal^=Sda. 

17.  Rq=q-^. 

25.  7a/?  =  -7/3a:. 

IS,  K(-q)==-Kq. 

26.  ap+^a=2Sa^. 

19.  Xa  =  -aa:. 

27.  ap-pa=2Vap, 

INTERPRETATION  OF  VECTOR  EQUATIONS  5^ 

28.  (a±/?)2=a2_t25a:/?+/?2. 

29.  Tipqr...)=Tp'Tq-Tr.., 

30.  U(pqr  .  .  .)  =Up-Uq-Ur  ,  .  . 

31.  S{pqr  .  .  . )  =S{qr  .  .  .  p)  =S(r  .  .  .  pg)  =.  .  . 

32.  Kipqr  ...)=...  Kr-Kq-Kp. 

33.  ;S(a:p+2/g  +  .  .  .  )''  =  xSp+ySq+.  .  . 

34.  V{xp-{-yq-\-.  .  .  )*=a:y7)+2/Fg  +  .  .  . 

35.  AS(g+r+s  +  .  .  . )  ^Sq+Sr-\-Ss^.  .  . 

36.  V{q^-r+s  +  .  .  .  )  =Vq-\-Vr  +  Vs+.  .  . 

37.  7g'g=>S5'7g4-^gFg'  +  7(F5'7^). 

38.  S{a'^r)  =Sa{Spr-^ypr)  =SaVpr' 

39.  SaVM^=SaMI^,  [M=m-\-pL. 

40.  =ASfa:  (m  +yO/?  =mSa^  ^S-aii^. 

41.  =mS^a-SP/ia.  [§  51. 

42.  =Spma-Sp}ia=S'pKMa.  [§122(5). 

43.  ^(a4-«)(6+/?)=.Sf(6+/?)(a+a:). 

44.  Spq=Sqp. 

45.  Kal^r=^^i^l^-r)  =Kr-Kap==-r-Pa. 

46.  aP-r-y-^a=2Sa^r  =  -2Srpa. 

47.  Sa^r=^ioL^-r)  S(P''d)  =Siq'P). 

48.  =AS(;'a5)  =>S;'«/?  =aS/?7'«. 

49.  ^(«./?.r)=^(>Sa:/?  +  Fa/?)r. 

50.  =>S.;'7a^  =  ->S.r7/?a. 
61.  =-,S-r(7/9a:+iS/?a). 

52.  =-SrPa. 

53.  ==-S.3ar. 

54.  iS(a:ia2.  •  .  a)n  =  (- l)«*S:(a'„  .  .  .  «2ai). 

55.  Sal3r=SaVl3r. 

56.  =>S-aF/?;'=^./?Fra:=ASfrFa/?. 

57.  a/?;'  +  ;'/?a=27a:/?;-=27;'^a. 

58.  2VaPr  =aPr -\-{ar^- arp-rap  +  raP)  +rl^a.  [57. 

*  See  Appendix. 


60  QUATERNIONS 

59.  2Vai^r=(^(l^r+rl^)-(^r+r(^)l^+r(^^-^M- 

60.  =2(aSpr-I^S(^r+rS(^l^)'  [26. 

61.  Vai^r-(^^^r=Va[^r-V'^^Pr' 

62.  =Va(^r-^M- 

63.  =V'aVpy. 

64.  =-pSra^-rSap.  [60. 

65.  =-F.7(/?r)«.  [62,25. 

66.  .-.  7-7(/?r)«=/5>Sf;'«-r>S^«/?.  [64. 

67.  7.(7a/?)7r^  =  -r^^7a:/?+^>S.(7a,/?)r.  [63,64. 

68.  =  -  r^apd  +dSa^r-  [^6,  24. 

69.  V'(Val^)Vrd=+aS-I^Vrd-l^S-{Vrd)a,  [66. 

70.  =aSl^r^-^Sard.  [56. 

71.  dSa^r  =(^S^r^  +^Srad  +  rSa^^.  [68,  70. 

72.  7-7a:/?7/?;'  =  ;'>S-(7a:^)/?-^>S:.r^«^.  [67. 

73.  =r^pVa^-pSra^.  [24,56. 

74.  =^Spar.  [^/?7a:^=0,     60. 

75.  a^a:/?;'  =  7-7/?a7Q:r.  [74. 

76.  ^Sapr=y-yrpyp^-  [48,  75. 

77.  rSa^r=vvarVr[^. 

78.  (5  =  -  (tiSi^  +/>Sy^  +A;aS/c^).  [71. 

123.  In  (63),  (64),  VaV^y  is  perpendicular  to  a  and 
coplanar  with  /?,  ;-.  That  it  is  perpendicular  to  a  can  be 
shown  by  multiplying  by  a  and  taking  scalars  thus, 

SaiV-aVpr)  =SarSa^-Sa^Sar=0.      [63,  64. 

It  is  in  the  plane  of  /?;-  since 

v-aV^r=r^<^?-^^(^r-  [63, 64. 

124.  In  (58),  (60)  Va^y  is  of  the  form  xa  +yl^-\-zy  and  is 
therefore  the  intermediate  diagonal  of  the  parallelepiped 
of  which  the  edges  are  aSl^y,  —^Say,  ySa^. 


INTERPRETATION  OF  VECTOR  EQUATIONS  61 

125.  In  (67),  since  S-rSal^=S'dSa[^=0, 

and  it  is  evident  that  V-  Vaj^Vyd  is  coplanar  with  d  and  y. 
Moreover,  since 

y •  Va^Vyd  =^V'VdrVap  =l^Sdra-aSpdr,     [25,  52,  67. 

it  is  also  coplanar  with  a  and  /?,  and  therefore  must  be  along 
the  intersection  of  the  planes  of  a,  /?  and  y,  d. 

126.  In  (71),  dSapy  is  the  intermediate  diagonal  of  the 
parallelepiped  of  which  the  three  edges  are  aS^yd,  j^Syad, 

rSapd. 


CHAPTER  XI 
QUATERNION  EQUATIONS   OF  THE  FIRST  DEGREE 

127.  An  equation  of  the  first  degree  with  respect  to  an 
unknown  quaternion  X,  is  that  which  contains  the  quater- 
nion to  the  first  power  together  with  known  quaternions, 
either  isolated  or  under  the  symbols  S  or  V. 

The  general  equation  will  then  have  the  form 

i:axb+i^csaxb+i.D'VaxB'E=^f, 

The  third  term  assumes  the  form  of  the  first  two  if  we 
replace  VAXB  by  AXB  —  SAXB,  so  that  the  general 
equation  reduces  to  the  form 

i:axb+^csaxb=f. 

128.  To  resolve  this  equation  we  decompose  the  quater- 
nions into  their  scalar  and  vector  parts.  Thus,  putting 
A=a+a,  B=h-\-^,  etc.,  we  have 

2(a+a)(:r  +  e)(6+/?)+S(c  +  ;')*S.(a+a)(a:+0(^+/?)=^  +  ^. 

The  sum  of  the  scalar  parts  of  this  will  be  found  to  be 
embraced  in  the  general  term  Sa^,  therefore  the  scalar  part 
of  the  equation  is /Sa^=c?. 

129.  Solution  of  the  linear  scalar  equation  Sa^=(i. 
This  may  be  written,  §  122,  (33), 

Sa{^-da-^)=0,  [ada  "i  =  d 

62 


QUATERNION  EQUATIONS   OF  THE   FIRST  DEGREE     63 

where,  §  56,  (2),  evidently  {$—da~^)  is  some  vector  /?,  JLa, 
or  p  =  VaY,  where  ;-  is  an  arbi- 
trary vector.     Therefore, 

The  geometrical  interpreta- 
tion of  this  is  shown  in  the 
diagram,  where  since  y  is  arbi- 
trary, the  locus  of  the  extremity  "^ 
of  ^  must  be  the  plane  _L  a  and 
through  the  point  da~^. 

130.  The  vector  part  of  the  second  term  in  §  128,  is, 
neglecting  the  S, 

Multiplying  out  the  first  term  we  find  the  vector  part  to  be 

axp-\-aM  -\-a^p-\-hxa  -\-xa^-\-ha^  +a^^. 
Of  these  forms 

=Sa^l3+aS^l3-^Sap+^Sae.      [§  122,  (61),  (64). 

Of  the  final  forms,  abx-y,  aS^^-y,  xSa^-y,  h-Sa^-y, 
Sa$l^-r,  ohxy,  ax^,  hxa,  aS$^,  xVad,  bSa^*  aS$^,  ^Sa^, 
are  comprehended  under  the  general  form, 

ah^,  $Sa^  are  comprehended  under  the  general  form, 
Vm$  =  V(ab+Sa^  +  .  ..); 


*  Or  neglected  as  purely  scalar. 


64  QUATERNIONS 

aF^/?,  bVa^  are  comprehended  under  the  form, 

Hence  the  vector  part  of  the  general  equation  becomes 

or  2a,S/?e  +  y(m  +  //)e=^.  [§  122,  (34). 

or  I,aS^e-\-V'Me=d.  [§61. 

This  is  generally  abbreviated  under  the  functional  sign, 

131.  Hamilton  called  this  a  linear  vector  function  of  the 
vector  ^.  Considered  as  an  operator  its  application  to  ^ 
has  many  interesting  results,  some  of  which  will  be  investi- 
gated in  the  following  pages. 

We  will  consider  first  the  properties  of  (j)  itself,  and  then 
the  result  of  its  application  to  the  vector  $. 

For  reasons  given  later  this  function  is  called  a  strain 
function. 

132.  Properties  of  ^.     Since  §  122  (35), 

and  §  122,  (36), 

V'M($  +  ri+...)=V'Me  +  V-Mr)+.,.  ; 
/.     I,l3Sa(e  +  7)+.  .  .  )  +V-M($  +  7)+.  .  .  ) 

=  (i:^Sa$  +  V-M$)  +(^pSa7)  +  V'M7))  -f .  .  .  , 
or  (t>(^  +  r)+.  .  .)=^e+^>?+.  .., 

that  is,  0  is  distributive  over  a  sum. 


QUATERNION  EQUATIONS  OF  THE  FIRST  DEGREE     65 

133.  li  $  =  r}=, , ,  to  n  terms, 

then  (fm^  =n(f)^ , 

that  is,  (j)  is  commutative  with  a  scalar  factor. 

134.  If  we  define  (j)~^  by  the  equation,  ^"^0=1,*  then 

and  from  §  132, 

or  e  +  >?  +  ...    =(?^-i(0^+0^  +  .  ..)• 

But  <j>^=d,  etc.,  and  (l)~^d  =  $,  etc.,  and  therefore, 

Hence  <f)~^  has  the  same  properties  as  (j). 
By  repeated  appHcations  we  can  easily  get,  for  all  integral 
values  of  k, 

135.  Conjugate   strain   functions.     Operating    on    ^f  = 

llaSp^  +  V'M^  with  >S-7?,  we  get 

=  ^S$l^Sarj+Srj(m  +  pL)^ 

=  ^S^l^Sa7)+S$im  +  n)r)  [§  122,  (24),  (38). 

=  j:S$^Sa7)+S$(m-fi)7j  [§  122  (52). 

=S${^l3Sar)  +  VKMt))  [§  122  (39) ,  (5) . 

4>  and  (j)'  are  called  conjugate  strain  functions.  They 
evidently  differ  in  the  interchange  of  the  known  vector  a, 
|9,  and  of  the  quaternion  M  and  its  conjugate  KM. 

*  See  Appendix. 


66  QUATERNIONS 

136.  When  (j>=(j)\  that  is,  the  function  conjugates  into 
itself,  the  functions  are  called  self-conjugate  strain  func- 
tions, in  which  case  St)^^  =S^(j)r). 

137.  Types  of  self-conjugate  functions.     Since 

and  writing  *  (cj)  +^0^  for  (l)^-\-^^^, 

which  shows  that  the  operator  </>  4-  9^'  is  always  self  con- 
jugate. 

138.  Furthermore,    S^^^  =Se(l>'^, 

whence  /Se(^-^0^=O, 

and  therefore 

the  vector  (^  —  (j)')  ^  is  perpendicular  to  I" 

or  (cl>-cj)0^==Ve$, 

where  e  is  some  unknown  vector. 

Consequently,    ^^  =  i  (^  +  ^0  ^  +  i  (9^  -  00  ^ 

which  shows  that  a  linear  vector  function  of  ^  differs  from  a 
self-conjugate  function  only  by  a  term  of  the  form  Vs^.  If  it 
is  already  self  conjugate  the  vector  £=0. 

139.  Since 

S'^cl>cl>^r)=S$ct>{ct>'7))  S-^cj>cj>'ri=S'-q<j>(^'^)      [§  135. 

=S'CJ>'T]ci>'^       [§135.  =S7i4>cj>'^, 

=S-<t>'^4>'ri\      [§122(24). 
therefore  ^^'  is  a  self-conjugate  strain  function. 

*  This  is  allowable  because  the  functional  symbol  <^  has  the  same 
properties  as  an  algebraic  factor  (distributive  over  a  sum  and  commu- 
tative with  scalar  factors)  and  can  be  treated  like  one.     See  Appendix. 


QUATERNION   EQUATIONS  OF   THE  FIRST  DEGREE    67 

140.  If  by  ((p-^g)^  we  understand  ^f +g6,  where  g  is  a 
scalar,  ^  +g  is  also  a  linear  vector  function,  since  it  has  all 
the  properties  of  0  as  the  reader  can  easily  demonstrate  for 
himself.     Hence 

S'^(cf>+g)r)=S-$(<j>7)-{-grj)  =S-^cl>7j-\-S'$g7), 

=S-i^ct>'^+9S'r)^  -S'rj(ct>^+g)e, 

or  <f>'  -\-g  is  a  conjugate  function  to  cj)  +g. 

141.  Application  of  <;6  to  a  vector  $,  If  an  elastic  soHd, 
that  is,  the  vector  connecting  the  several  elements,  be 
subjected  to  the  operation  ^,  then 
all  its  particles,  for  instance,  those  /  \pa 
determined  by  the  vectors  a,  p,  y,  are 
displaced  to  positions  determined  by  /  J^ 
the  vectors  (j)a,  0/?,  0;'.  In  general, 
any  particle  whose  vector  is  ^  occupies  ^ 
after  the  operation  the  position  whose  vector  is  ^^. 

Also  any  vector  a  is  displaced  to  the  position  ^a,  for  since 

^a  =(j)Y  —  <j)^. 

Hence,  Any  straight  line  of  particles  parallel  to  a,  ^ay  xa, 

is  homogeneously  stretched  and  turned  by  the  operation  cj)  into 

a  straight  line  of  particles  parallel  to  ^a,  and  the  ratio  of 

.    X(ba     (f)a 
extension  is  —^~-  =  — . 
xa        a 

The  operation  (j>  is  called  a  strain  and  the  property  that 

parallel    lengths    are    strained    into    parallel    lengths    and 

stretched  proportionally  is  the  physical  definition  of  linear 

homogeneous  strain.    Portions  of  the  body  originally  equal, 


68  QUATERNIONS 

similar  and  similarly  placed  remain  after  the  strain  equal, 
similar  and  similarly  placed. 

142.  If  in  the  general  equation  of  the  first  degree  with 
respect  to  an  unknown  quaternion,  §  127,  instead  of  sepa- 
rating the  scalar  and  vector  parts,  we  had  merely  indicated 
the  vector  part,  thus 

I,VQXR  =  VF, 

this  operator  VQ(  )R  must  of  course  be  the  undeveloped 
form  of  ^. 

If  R=Q~^,  we  have  the  rotator  of  §  102  as  a  special  case 
of  0. 

If  Q  and  R  degrade  into  scalars,  we  have  SFmX=nX, 
or  dilatation  merely. 

A  combination  of  rotation  and  dilatation  makes  the  strain 
just  defined. 

143.  Properties  of  ^.  To  get  a  slightly  different  view, 
let  us  suppose  «,  /?,  7-  to  be  unit  vectors  at  right  angles  to  each 
other.     Hence,  §  122  (78), 

By  the  definition  of  homogeneous  strain  this  is  changed  into 

where  a\  /?',  f  are  three  vectors  upon  which  the  same 
proportional  distances  Sa$,  S^^,  Sy^  are  laid  off.  This  is 
necessary  in  order  to  preserve  the  similarity  required  by 
homogeneous  strain.     By  substituting  a,  p,  f  for  ^,  we  find 

«'=0«,     /?'=#,     r'=#. 


QUATERNION  EQUATIONS   OF  THE  FIRST  DEGREE    69 

But 

^a  =«'  =  -  {aSaa'  +^S^a' +rSra')  =aA  „  -\-l^Ba  +  rCa, 

^^  =^'  =  -  {aSap'  +/?^/?/?'  +  r^rP')  =aA  ^  -V^Bp  +  yC ^, 

^r  =  f  =  -  (<^Saf  +l^S^f+ySrf)  =aAr  +^B,  -{-jCr, 

which  equations  contain  nine  arbitrary  constants,  Saa\ 
S^a',  etc.,  since  a',  /?',  f  are  entirely  independent  of  a, 

.8,  r- 

Operating  on  (f)^  with  S-t)  we  have 

5.  ,^e  =  -  (Sa'fjSa^  +Sl^'7)S^^  -}-SfrjSr^) 

=    -S-^(aSa'rj+^dSl^'r)+rSfr)),  [§122(24). 

where  the  expression  in  the  parenthesis  is  a  linear  vector 
function  of  tj,  which  will  be  shown  (§  144)  to  depend  upon 
the  same  nine  scalars  il«,  B^,  etc.,  as  those  in  </>,  and  which 
we  may  therefore  appropriately  designate  by  ^',  thus 

whence  obviously        S-i^^$  =*S-  <^^')^, 
as  before. 

144.  Substituting  «,  /?,  ;-  for  tj  in  the  expression  for  (^'t^, 
we  find 

4>'r=^-{aSa^r'rps?'r-\-rSrr), 

or  using  the  notation  of  §  143, 

<^'a:  =  a  A„  +/?^  p  +  T-^r  i 


70  QUATERNIONS 

which  shows  that  ^'  depends  upon  the  same  nine  scalars 
as  (^. 

145.  If  a,  /?,  Y  are  given  non-coplanar  vectors,  then  §  29, 

whence  (j)^  =X(j)a  +y<f>[^  +z(l>]r. 

The  vector  <^6  is  known  when  the  three  vectors  (jya,  ^/?, 
(py,  are  known.  Each  of  these  vectors  involves  three  scalar 
constants  as  in  the  case  of  $,  multiples  of  the  reference 
vectors  a,  /?,  y.  Therefore  the  value  of  0  depends  upon 
nine  scalar  constants.  It  has  therefore  been  called  a 
nonion. 


CHAPTER  XII 

APPLICATIONS   OF  ^ 

146.  Changes  of  volume  due  to  <^.     p,  c/S^,  cfy^p  are,  in 
general,  not  in  one  plane,  and  hence,  §  29, 

<p^p=m2(l)^p  —  mi^p+mp,      .     .     .     .     (1) 

where  m2,  mi,  m  are  scalars  independent  of  p.  The  inde- 
pendence is  obvious,  since  we  may  put  «,  /?,  ^  in  succession 
for  p  and  thus  obtain  three  equations  from  which  they  can 
be  obtained. 

From  §  59  the  volume  of  the  parallelopiped  whose  three 
conterminous  edges  are  p,  <j)p,  <pp  is 

—  S'p<j)p(jy^p. 

After  the  strain  this  volume  is 

—  S  •  (j)pc[)^p<j)^p. 

Multiplying  Eq.  (1)  by  (j)p<j)^p  and  dividing  by  pcjypcjy^p,  we 
have 

S'(j)p^'^p4>^p 
S'pcj>pct>^p    ""^ ^^^ 

This  ratio  of  dilatation  m  due  to  ^,  the  ratio  between  the 
volumes  after  and  before  the  strain,  is  called  the  modulus 
of  the  strain  <j). 

71 


72  QUATERNIONS 

147.  By  referring  to  the  equations  of  §  143  we  see  that 
the  scalar  part  of  the  product  <f)a(j>P(j)y  is  confined  to  those 
terms  in  which  all  three  of  the  vectors  a,  p,  y  appear,  that 
is,  taking  S-a^y  =  l, 

'^~   s-apr 

^CMpBr-ArB^). 
In  the  same  way  from  §  144, 

=m. 

Hence,  Conjugate  strains  produce  equal  changes  of  volume. 

148.  Special  values  of  m.  If  m  =  1  there  is  no  change  of 
volume  caused  by  the  strain. 

If  m=0,  there  results  a  zero  volume  due  to  the  solid 
becoming  strained  into  a  plane,  a  line,  or  a  point,  and  in 
this  case  cf)  is  called  a  null  function,  singly,  doubly,  or  triply 
null  according  as  the  strain  results  in  a  plane,  a  line  or  a 
point.  The  plane,  line,  or  point  into  which  the  solid  is 
strained  is  called  the  strain  plane  of  (j),  strain  line,  etc. 

When  m=0,  then  S(j)a(j)p^y ,  §  147,  becomes  zero  and, 
§  60,  (j)a,  <j}pj  (jyy  are  coplanar  vectors,  and  we  can  have  a 
relation, 

or  §  132,  cl){xa  ^y^-^zf)  =0. 

The  vector,  xa+yl^-\-zj-,  whose  strain,  i.e.,  the  result  of 
the  application  of  the  strain  function  (p,  is  zero  is  called  a 
null  direction  of  (f>.  In  this  case  ^  is  singly  null,  unless 
(f>a,  (j)^,  (j)Y  becomes  collinear. 


APPLICATIONS  OF   cj)  73 

149.  If  there  is  only  one  vector,  say  a,  whose  strain  is 
zero,  i.e.,  (fta  =0,  then 

and  therefore  m=0,  and  ^/?  cannot  be  parallel  to  (fyy 
(i.e.,  ^^f^xcjyy),  for  otherwise  (j)(^  —  xf)  would  be  equal  to 
zero  and  there  would  be  a  second  vector,  ^  —  xy,  with  a 
strain  of  zero,  which  is  contrary  to  the  hypothesis.  Hence, 
if  there  is  only  one  null  vector  and  p=xa  •\-yp-\-Z'f,  then 

<j)p=ycl>l^-{-z<l)r, 

and  (jyp  is  a  vector  In  a  plane  parallel  to  ^,/?,  <^;',  that  is,  <^ 
is  singly  null. 

150.  If  there  are  two  (only)  vectors  whose  strain  is  zero, 
say  ^a:=0,  (f>l^  =0,  then  (j)p=Z(f>y,  which  confines  <j)p,  the 
strain  of  the  general  vector  p,  to  the  line  parallel  to  (j))-, 
and  in  this  case  ^  is  doubly  null. 

151.  Similarly,  if  <;6a  =0,  #=0,  cjyj-^^O,  then  ^^=0  for 
all  values  of  p  and  (^  is  a  triply-null  function. 

152.  When  ^  is  singly  null,  say  (j)a  =0,  then  the  general 
vector  of  any  point  in  the  strain  plane  of  (j)  is,  §  149, 

<j)p=ycl)^+Z(j)r, 

and  all  particles  that  strain  into  this  plane  have  the  vectors, 

p=xa+yl^+z-jr, 

where  x  is  arbitrary,  since  <j)a  =0,  and  therefore  the  locus 
of  |0  is  a  line  parallel  to  a.     Hence, 

A  singly  null  function  strains  each  of  its  null  lines  into  a 
definite  point  of  its  strain  plane.     Similarly, 

A  doubly-null  funcion  strains  each  of  its  null  planes  into 
a  definite  point  of  its  strain  line. 


74  QUATERNIONS 

153.  Special  applications  of  ^.     What  vectors,  if  any,  are 
unchanged  in  position  by  the  strain  (f>f 
If  p  is  unchanged  by  the  strain,  then 


(jyp 

^S'iO, 

(j)2p^g2p^ 

cj^^p: 

=9\ 

and  the  equation 

of  §  146  becomes 

(.7^- 

-m2g^-\-mig-'i 

m)p=\ 

[),     . 

.  .  (1) 

which  must  have  one  real  root  gi  and  may  have  three.  In 
other  words,  the  curious  fact  that  however  a  body  may  be 
homogeneously  strained,  there  is  always  at  least  one  vector 
whose  direction  remains  unchanged.     Hence 

and  S-  /i(j)p  —  giS/xp  =0,  [/i=any  vector, 

and  S-p{cl>'ii-g,/i)=0=S'p{cl>'-g)/L  [§  135. 

Hence,  §  56  (2),  (^'  —  gfi)/z  is  perpendicular  to  ^o. 

From  this  it  appears  that  the  operator  {(j)'  —  g)  applied  to 
any  vector  /z  throws  it  into  a  plane  perpendicular  to  p,  or, 
in  other  words,  cuts  off  the  component  of  the  strained  fx 
which  is  parallel  to  p. 

Also,  p\\y-{<t>'-Qi)P-^^'-gi)^,  [/I,  V  any  vectors, 

or  multiplying  out 

^||7(^V(/)'y-^i(r/>>.i;  +  /i0^) +sriV) 

154.  Properties  of  i^'  —  g).  Like  0,  this  operator  pro- 
duces a  homogeneous  strain,  as  can  be  shown  in  a  manner 
similar  to  that  in  the  case  of  (j).  It  has  the  same  general 
properties  and  has  a  modulus  nig.     Omitting  the  primes, 


APPLICATIONS  OF  ^  75 

which  will  not  affect  the  discussion,  we  have,  after  expand- 
ing, 

155.  In  §  153  we  found  the  expression, 
V-{ct>^-gi)fx{cl>'-gi)v, 

the  vector  part  of  the  product  of  two  strained  vectors.  To 
investigate  this  we  return  to  the  last  equation  of  §  146, 
which  can  be  written, 

_s-4,a4,p4>r   s-{4>?<j>Ma  _8a<t,'V{<i>^<i>r)    . .  ,„, 

or  since  a  is  any  vector  whatever, 

^'7(#(jSr)  =m7/?r, (1) 

or  since,  §  147,  cj)  and  ^'  have  the  same  modulus, 

<l>v(<i>Wr)  =mv^r, 

Using  (<j)  —  g)  in  place  of  <f>,  we  have 

(4.-g)V(4>'  -g)p{^'  -g)r=m„v^r 

={<p-g)  V[<l>'^4>'r  -9(4''^  ■  r  +P<P'r)  WM 
-{4>-  g)[v<j>'^4,'r-g{v<i>'^  ■  r + vp<i>'r)  +g^vpr\ 
^{ci>-g)[m4>-^v^r-9(.v<j>'^-r+v^'t>'r)  +9^vpr]  [d) 

=={m-in'ig+m'2f-g^)VPr- 


76  QUATERNIONS 

156.  In  this  equation  and  the  last  expression  of  §  153,  we 
have  the  four  vectors  Vj^y,  V-^'P-y,  V-^<j)'x,  (jiVPy,  and 
we  can  write 

Operate  successively  with  S-a,  S-^,  S-y  and  we  get 

spr^y^zs-rpct>y. 

But  §  122  (53),  y  =  -l,z  =  -l,and 

X  =—^^n^ ^L-^ 1^:^^!.  =.rn'2.       [§  154. 

157.  Hence,  §  156, 

or  substituting  in  the  equation  of  §  155  and  using  $  for 
VPr,  we  get 

or  expanding, 
{m+gcj)^+m'2g<l>+g^(t>-g'Jn^~'^+g^m'2-g^-g^)^ 

=  {m-m\g+m'2g^-g^)^, 
or  {ct>'^-m'2(j>-m^-^)^^-m\^,       .     .     .     (1) 

or  {(j)^  —  m'2(l>^+rn'i^  —  m)$=0, 

where  ^  is  any  vector  whatever. 


APPLICATIONS   OF  ^  77 

158.  Comparing  this  with  Eq.  (1),  §  146,  we  have 

The  expression  of  §  153  now  becomes,  by  §§  155,  157, 
and  the  equation  above, 


^||(m0-i-gfi(m2-^)+gfi2)e 


[dividing  by  gi  and  operating  with  ^  or,  multiplying  and 
dividing  by  ((f>  —  gi)  and  then  substituting  the  value  of 


9i 


from  (1)  of  §  153, 


p\\ ^^ 6.        [6  =  any  vector. 

159.  Since  (1)  of  §  153  is  true  for  all  vectors,  and  since  0 
is  commutative  with  scalars,  it  can  be  written, 

and,  §158,  ^||(^_^2)(<5^-^3)e, 

that  is,  the  operator  (^— S'2)(^  — S^s)   when  applied  to  any 
vector  whatever  results  in  a  vector  parallel  to  p,  where 

{cj>-g{)p==0, 

or  <j>p=g\p, 

p  being  unchanged  in  direction  by  the  strain  (j>. 


78  QUATERNIONS 

160.  We  can  fortify  this  by  another  line  of  attack.  As 
before,  §  153,  under  the  condition,  ^^  =g^,  we  have 

Supposing  for  the  moment  that  the  three  roots  are  real, 
the  solution  of  the  problem  will  be  given  by  one  of  the 
directions  a,  p,  y  which  satisfies  the  conditions, 

{cj>-g{)a=0,     (0-^2)/?=O,     (0-^3)/9=O.      .     (1) 

Now  ^=aa-{-hp-\-cj. 

Operating  with  {(j)  —  gi),  we  get 

(ct^-gi)^=b{ct>-gi)^-\-c(cf>-g,)r 

=  {b(j)-hgi+bg2-b(t))^-\-{ccl)-cgi-\-cgs-ccf))f 

[subtracting  h{^  —  g2)^=0     and    c{(j)  —  gs)}'=0 

=Hg2-gi)P +c(g3-gi)r' 

161.  Thus  we  see  that  the  operator  {4>—g\)  cuts  off  from 
the  general  vector  ^  the  component  parallel  to  a. 

Operating  again,  with  (^— §^2),  this  becomes 

^<f>-gi){4>-g2)^ =c{g3-gi){g3-gi)r' 

In  the  same  way, 

((t>-gi)((l>-g3)^=b(g2-gi)(g2-g3)P, 
((l>-g2)i(l>-g3)^=aigi-g2)(gi-g3)a. 
That  is,  the  operator, 

((l>-g2){(j>-g3), 


APPLICATIONS  OF  (f>  7^ 

operating  on  a,  since  $  is  any  vector,  leaves  its  direction 
unchanged,  and 

«ll(<5^-9'2)(^-g'3)^. 

If  the  three  roots  gi,g2, 93  are  unequal  the  three  unchanged 
directions  are  given  by  the  operators  above. 

162.  Those  lines  that  are  unchanged  in  direction  by  the 
strain  <f>  have  been  suggestively  called  latent  lines  of  ^,  and 
evidently  from  the  relation  {(t>  —  gi)p=^,  the  latent  lines  of 
0  are  the  null  directions  of  (<f)—gi). 

Hence  ^  — gri  is  a  null  strain,  or  §  154, 

mod.  (^  —  gi)=mg=m  —  mig+m2g^—g^=0. 

The  roots  of  this  cubic,  gi,g2,  gs  are  the  ratios  of  dilatation 
or  extension  of  the  latent  lines  of  (j)  and  are  called  the 
latent  roots  of  <^.  Planes  whose  vectors  are  not  strained  out 
of  the  plane  by  0  are  called  the  latent  planes  of  <j). 

163.  If  a,  ^,  y  are  the  latent  directions  of  cj),  with  the  latent 
roots  gi,  g2,  gz,  then  (/?,  ;-),  (;-,  a),  (a,  /?)  determine  the  latent 
planes  of  (p,  which  are  enlarged  by  the  strain  (j>  in  the  ratios 

g2gz,  g^gi,  gm- 

For  if  x^  +2/7'  be  a  vector  in  the  plane  /?;-,  then 

cj>  {xp +yr)=  a:# +y(j>r'-  ^g2P + ygz  r, 

and  evidently  the  strained  vector  remains  in  the  plane, 
though  its  direction  has  been  changed.  q.e.d. 

Also  for  the  two  vectors  in  the  plane  /?,  ;', 

T7^(x/3  +yr)  (xip  +yir)  =  TV{xg2^  +ygsr)  (xm^+yigsr) 

=g2g3TV(xp +yr)(xip +yir.)  q.e.d. 

Conf.  §  58.] 


80  QUATERNIONS 

164.  If  92=93,  and  we  operate  upon  upon  ^  (§  160)  with 
(j>  —  92,  remembering  that  {^  —  92)P  =  {^  —  92)T'^^}  we  get 

{4>-92)^=a{gi-g2)a, 

and  the  latent  direction  is  given  by 

Operating  upon  ^  by  {(j>—gi)  we  get 

{<t>-9i)^  =  i92-9i){hP+cr), 

Since     (96-9^2)  W+cr)  =(#-^2/?)  +c(^-^2)r, 

=0       [(^-9r2)/?=0,  {cjy-92)r=0. 

therefore  every  line  of  the  plane  (^ly  is  unchanged  in  direc- 
tion, as  well  as  kept  in  the  plane. 

165.  gi  =02  =93- 

Operating  on  ^=aa  +6/?+cgr  with  4^  —  91  we  obtain 

(0-Sri)e=O, 

that  is,  when  the  latent  roots  are  all  equal,  all  the  vectors  are 
latent  vectors. 

166.  There  cannot  be  two  latent  directions  for  the  same 
root  9i.  For  if  ((j)—9i)d=0  as  well  as  {(f)  —  gi)a=0,  we 
should  get  by  the  method  of  §  161, 

{(l>-92){<j>-93)^=d(gi-g2){gi-g3)^, 
or  a{gi-g2)  (gi  -93)0^  =d{9i  -92)  {91-93)0, 

which  cannot  be  unless 

9^1  =92,      9i  =93r 
that  is,  unless  the  roots  are  all  equal. 


APPLtCATlONS  OF  ^  81 

167.  Since  the  latent  planes  are  strained  in  different 
ratios  (the  latent  roots  being  unequal)  they  cannot  coincide 
and  their  intersections,  the  latent  vectors,  a,  /?,  7-,  cannot  be 
coplanar. 

^  —  g\  strains  all  vectors  into  the  plane  ^,  y,  that  is,  into 
the  plane  determined  by  the  other  latent  vectors. 

For  if  ^  =xa+y^+zy  =  general  vector, 

then  {^-g\)^=x{<j)-gi)a+y{<h-gi)P+z{cj)-g{)y 

=yig2-gi)P+z(g3-gi)y,    [(0-gfi)o:=O. 

=vector  in  plane  p,  y.  q.e.d. 

Repeating  this  operation  we  get 

(^-^1)  (0-^2)  (^-6^3)6=0, 

as  already  shown  in  §  159. 

The  strain  plane  of  (j)  —  gi  is  the  latent  plane  of  cj). 

168.  Two  conjugate  strains  have  the  same  latent  roots.     For 

since  {(j)  —  gi)a=0. 

.'.    0=S^{cl,-g,)a=Sa(cl>'-g^)$,  [§135. 

and  evidently  (t>'  —  gi  strains  all  vectors  into  a  plane  per- 
pendicular to  a,  i.e.,  (f>' —gi  is  a  null  function  and  gi  is  a 
latent  root  of  ^'  (§  162). 

169.  The  latent  plane  of  one  strain  is  perpendicular  to  the 
corresponding  latent  line  of  the  conjugate  strain. 

The  strain  plane  of  (j>^  —  gi  is  the  latent  plane  of  ^',  §  167, 
and  therefore  the  latent  plane  of  0'  is  perpendicular  to  a, 
the  latent  line  of  <j). 


82  QtTATERNIONg 

170.  //  a  strain  is  self  conjugate  its  three  latent  roots  are 
real. 

Suppose  Qi+ti^  —  I  to  be  one  of  the  roots,  and  let  the 
application  of  the  other  factors  of  the  cubic  be  denoted  by 

Then,  by  §  160(1), 

^(«  +aiV~l)  =  {gi  -{-tiV^)  (a  +ai\/^, 
or  equating  the  reals  and  imaginaries, 

(pa^gia  —  tiai,       (l)ai=giai+tia. 
Whence  Saicpa^giSaia  —  tiSaiai, 

/Sa^rti  =giSaai  -\-tiSaa, 
or,  since  Sai(f)a=Sacj)ai, 

0=t,{a^+ai^), 
whence  ti  =0.  q.e.d. 

171.  If  a  strain  is  self  conjugate,  that  is,  if  ^=0'  or 
Srj^^=S$(l)7),  then  by  §  170  it  must  have  three  mutually 
perpendicular  latent  directions. 

Conversely:  If  ^  have  three  mutually  perpendicular 
latent  directions  i,  j,  k,  with  the  corresponding  latent  roots, 
a,  b,  c,  then  cj)  is  self  conjugate.     For,  §  122  (78), 

$  =  -iiSie+jSj^+kSm, 

(j)^  =  —  (aiSi^  -\-bjSj^  -{-ckSk^  ;         [(j>i  =ai,  etc. 

S^(l>$  =  -  aSrjiSi$  -  bSrjjSj^  -  cSfjkSk^; 


APPLICATIONS  OF  (fy  83 

7)  =  —  (iSi7)-\-]'Sjrji-kSk7j); 
^T}  =  —  aiSiTj  —  bjSJT)  —  ckSkr) ; 
S$(l)7)  =  -  aSi^SiT}  -  bSj^Sjrj  -  cSk^Skrj 

and  (j)  is  self  conjugate.  q.e.d. 

172.  When  the  strain  is  self  conjugate  (0=^')  and  the 
latent  roots  equal,  the  strain  is  non-rotational  and  is  called 
pure. 

Since  the  strain  is  self  conjugate,  §  168,  the  latent  lines 
are  i,  /,  k,  and 

$  =  -iSi^-jSj^-kSk^, 
(f)^  =  —  aiSi^  —  ajSj^ — akSk$ 
=a(-iSi^-jSj^-kSk^) 
==a$,  [a  =  latent  root. 

and  ^  is  not  rotated. 

That  is,  all  vectors  are  latent  lines.  q.e.d. 

When  ^  is  self  conjugate  ((j)  —  (j)'),  by  reference  to  §§  143, 
144,  if  the  strain  is  given  by 

(j)i=xi-{-yj+zk; 

(j)j=x'i-\-y'j-\-z'k', 

(j>k=x"i^-y''i+z''k\ 

then  when  the  strain  is  pure, 

x'=y,       x"=z,       y"=z', 

and  a  pure  strain  depends  upon  six  instead  of  upon  nine 
scalar  constants. 


84  QUATERNIONS 

173.  //  the  drain  is  self  conjugate  (<f)  =  0')  and  the  three 
latent  roots  a,  b,  c  are  not  equal,  only  the  vectors  i,  j,k  are  non- 
rotated. 

As  in  the  previous  section, 

^e  =  -  aiSi^  -  bjSje  -  ckSk^ 

7^X^.  Q.E.D. 

<f)i  =  —  aiSii  —  bjSij — ckSik 
=ai. 
Similarly,  ^/=  6/,     ^k=ck.  q.e.d. 

174.  Example.  If  <j)  have  three  latent  directions  a,  p, 
y  with  the  three  latent   roots  a,  b,  c,  then 

^^= — ^^^r~' 

and  <f>=<i>',  or  <}>  is  self  conjugate.     Where  is  the  error? 

Ans.  (j)'a7^aa. 

176.  Examples.  Solve  Vae^  =  r  =  <t>^  for  e.  By  §  146 
(2), 

O  •  A/IV 


APPLICATIONS  OF  0  85 

where  X,  n^  v  are  any  three  non-coplanar  vectors.     Putting 
for  these  a,  ^,  y,  if  they  are  not  coplanar,  we  get 

[VarP  =aSrP-rSa^  +^Sar.      [§  122  (64). 
By  §  158, 

m2  =  —  Sa^. 
By  §  157(1),  we  get 

[§  122  (64). 
whence  ^  V  =  "^= — ' SB      


APPENDIX 


FUNCTIONAL    SYMBOLS 

As  every  discrete  magnitude  is  of  necessity  derived  from 
unity  by  some  algebraic  operation,  so  the  symbol  repre- 
senting a  discrete  magnitude  can  be  considered  as  the 
symbol  of  some  operation  whose  operand  is  unity. 

Symbolizing  the  operation  of  converting  unity  into  the 
magnitude  x  by  the  symbol  x-1,  we  have  the  functional 
equation  {x  operating  upon  1), 

X'l  =x. 

A  second  application  gives 

x{x-l)  =x'^-\=x'^, 

where  the  2  of  x'^-l  shows  the  number  of  operations,  and 
the  2  of  x^  the  result. 

The  inverse  of  the  operation  must  be  symbolized  by 

X 

In  the  expression  x-1  occur  two  concepts,  the  operation^ 
X'l  and  the  effect  of  the  operation,  x. 

87 


88  QUATERNIONS 

The  inverse  of  the  operation  is  x~^-l= — 1,  the  inverse 

of  the  effect  is  (x)~^=— ,  and  in  this  case  the  two  results 

coincide. 

If  we  take  a  different  operand,  say  y,  then  x~^-y  and 
{^y)~^i  the  inverse  of  the  operation  and  the  inverse  of  the 
effect  are  not  the  same. 

Every  algebraic  expression  can  be  considered  as  a  func- 
tional operation.  Thus  adding  1  to  x  can  be  considered  as 
an  operation  symbolized  thus, 

/(a;)=x  +  l. 

The  inverse  operation  would  be  whatever  operation  is 
necessary  to  change  back  from  x  + 1  to  a;  to  original  operand. 
Thus 

Here  again  /"^t^t,  where,  of  course,  f~^  symbolizes  the 

inverse  of  the  operation  and  (f)~^  ^j,  the  inverse  of  the 
effect. 

Similarly,  log~i  x  ^  (log  x)  ~i, 

sin~i  X7^(sin  x)"^. 

Examples.     If/(x)=x  +  l,     f{x)^x-\,    p{x)=x^-2. 


If    /(x)4+.,  /-Kx)=--=^,   /.(.)=-^+i±^. 


APPENDIX  89 

If  f(x)  =x2  +2a;  +6,      /"i  {x)=-l±  V^T, 

P(x)  =  (x2  +2x  +6)2  +2(a;2  +2x  +6)  +6. 
In  these  examples,  omitting  the  operand,  show  that 

In  the  same  way  prove 

sin  sin~i  x  =x^  log  log~^  x  =x, 

log~^  log  x=x,  sin~i  sin2a;  =sin  x. 

If  fix)  =x^+3,      F{x)  =2-V^,  then 


fF(x)=(2-\  x)^+'S,        F'f{x)=2-Vx^-^S. 
2x—\ 

The  algebraic  symbol  is  distributive  over  and  commuta- 
tive with  its  operand,  that  is,  xy+xz  +  . .  .=x{y+z+, , .  ), 
xy=yx. 

Other  symbols  of  operation  which  like  these  are  dis- 
tributive over  and  commutative  with  the  operand  will  be 
subject  to  the  same  algebraic  laws.  Hence  we  can  treat 
these  symbols  of  operation  just  as  we  treat  algebraic  symbols 
of  operation. 

Hence  we  can  write  §  122  (33),  (34),  §  132,  §  135, 

Sp  +  Vp  =  (S  +  V)p;  ct>^-cj>'^  =  {^-cf>')^, 


90  QUATERNIONS 

We  recall  the  similar  results  in  Calculus,  where,  omitting 
the  operand,  we  have 

(D2-a2)=(Z)+a)(Z)-a)  [0=^ 


dx 


(D2-D-2)=(D  +  l)(D-2) 


xW2  =x^D-D=  x^D-  [xD  =  0. 

X 


=  0.0-6  =  6(0-1), 

where  0  is  commutative  with  constants  but  not  with  either 
X  or  D. 


INDEX 


Addition,  3 

Algebra,  31 

Amplitude,  7 

Angle  of  a  quaternion,  39 

Argand  diagram,  7 

Arithmetic,  30 

Axis  of  quaternion,  21 

Calculi,  2 

Complex  functions,  31 

Complex  quantity,  7 

Conjugate  of  a  quaternion,  33 

Conjugate-strain  function,  65 

Continuum,  1 

Coplanar  quaternion,  46 

Diplanar  quaternions,  36 

Discreta,  1 

Division,  4 

Euler*s  Theorem,  28 

Evolution,  5 

Functional  symbols,  87 

Harmonic  displacement,  11 

Idiograph,  6 

Idiograph,  space,  10 

Involution,  4 

Latent  lines,  79 

Latent  planes,  79 

Linear  homogeneous  strain,  67 

Linear  scalar  equation,  62 

Mean  reversion,  5 


Modulus,  7,  71 
Multiplication,  4 

*  *  of  vectors,  17 

Nonion,  70 
Null  direction,  72 
Null  function,  72 
Operation  A,  B,  11 
Opposite  quaternions,  33 
Parallel  vectors,  17 
Plane  of  quaternion,  21 
Plane  of  a  vector,  21 
Quadrantal  versors,  50 
Quaternion,  21,  31 

"         ,  angle  of,  39 

'*         ,  axis  of,  21 

**         ,  plane  of,  21 

**         ,  reciprocal  of,  32 
Quaftemions,  equality  of,  35 

"  ,  diplanar,  36 

Revector,  12 
Reversion,  3 
Rotator,  52,  68 
Scalar,  9 
"      part,  27 
*  *      of  the  unit  part,  27 
Self-conjugate  strain  function,  66 
Singly  null,  72 
Space  idiograph,  10 
Stroke,  6 

91 


92 


INDEX 


Strain,  67 

Strain,  linear  homogeneous,  67 

Strain  function,  64 

"  "        conjugate,  65 

Strain  plane,  72 
Subtraction,  4 
Tensor,  13,  27 

**       of  the  vector  part,  27 
Unit  part,  27 

"     vector,  13 

"         "      of  the  vector  part,  27 


Vector,  U 

"      ,  addition,  etc.,  12 
unit,  13 

'  *      ,  inclined,  18 

"      ,  parallel,  17 

' '      ,  perpendicular,  18 

"       part,  27 

' '       part  of  the  unit  part,  27 

"      ,  plane  of,  21 
Versors,  48,  49 

"      ,  quadrantal,  50 


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Vol.   I.     The  Generating  Plant 3  00 

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Crocker,  F.  B.,  and  Arendt,  M.     Electric  Motors Svo.  *2  50 

Crocker,  F.  B.,  and  Wheeler,  S.  S.     The  Management  of  Electri- 
cal Machinery i2mo,  *i  00 


8      D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Cross,  C.  F.,  Bevan,  E.  J.,  and  Sindall,  R.  W.     Wood  Pulp  and 

Its  Applications.  (Westminster  Series) 8vo  {In  Press.) 

Crosskey,  L.  R.     Elementary  Prospective 8 vo,  i  oo 

Crosskey,  L.  R.,  and  Thaw,  J.     Advanced  Perspective Svo,  i  50 

Davenport,  C.     The  Bookj^    (Westminster  Series.) Svo,  *2  00 

Da  vies,  E.  H.     Machinery  for  Metalliferous  Mines Svo,  S  00 

Da  vies,  D.  C.     Metalliferous  Minerals  and  Mining Svo,  5  00 

Davies,  F.  H.      Electric  Power  and  Traction Svo,  *2  00 

Dawson,  P.     Electric  Traction  on  Railways Svo,  *9  00 

Day,  C.     The  Indicator  and  Its  Diagrams i2mo,  *2  00 

Deerr,  N.     Sugar  and  the  Sugar  Cane Svo,  *3  00 

Deite,  C.     Manual  of  Soapmaking.     Trans,  by  S.  T.  King.  .4to,  *5  00 
De  la  Coux,  H.     The  Industrial  Uses  of  Water.     Trans,  by  A. 

Morris Svo,  *4  50 

Del  Mar,  W.  A.     Electric  Power  Conductors Svo,  *2  00 

Denny,  G.  A.     Deep-Level  Mines  of  the  Rand 4to,  *io  00 

Diamond  Drilling  for  Gold *5  00 

Derr,  W.  L.     Block  Signal  Operation Oblong  i2mo,  *i  50 

Desaint,  A.     Three  Hundred  Shades  and  How  to  Mix  Them.  .Svo,  *io  00 

Dibdin,  W.  J.     Public  Lighting  by  Gas  and  Electricity Svo,  *S  00 

Purification  of  Sewage  and  Water Svo,  6  50 

Dietrich,  K.     Analysis  of  Resins,  Balsams,  and  Gum  Resins  .Svo,  *3  00 
Dinger,  Lieut.  H.  C.     Care  and  Operation  of  Naval  Machinery 

i2mo.  *2  00 
Dixon,  D.  B.     Machinist's  and  Steam  Engineer's  Practical  Cal- 
culator   i6mo,  mor.,  i  25 

Doble,  W.  A.     Power  Plant  Construction  on  the  Pacific  Coast. 

{In  Press.) 
Dodd,  G.     Dictionary  of  Manufactures,  Mining,  Machinery,  and 

the  Industrial  Arts i2mo,  i  50 

Dorr,  B.  F.     The  Surveyor's  Guide  and  Pocket  Table-book. 

i6mo,  mor.,  2  00 

Down,  P.  B.     Handy  Copper  Wire  Table i6mo,  *i  00 

Draper,    C.    H.     Elementary   Text-book    of    Light,    Heat   and 

Sound i2mo,  i  00 

Heat  and  the  Principles  of  Thermo-dynamics . . . i2mo,  i  50 

Duckwall,  E.  W.     Canning  and  Preserving  of  Food  Products 

Svo,  *5  00 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG      9 

Dumesny,  P.,  and  Noyer,  J.     Wood  Products,  Distillates,  and 

Extracts 8vo,  *4  50 

Duncan,  W.  G.,  and  Penman,  D.     The  Electrical  Equip  nent  of 

Collieries 8vo,  *4  00 

Duthie,    A.    L.     Decorative    Glass    Processes.     (Westminster 

Series) 8vo,  *2  00 

Dyson,  S.  S.     Practical  Testing  of  Raw  Materials 8vo,  *5  00 

Eccles,  R.  G.,  and  Duckwall,  E.  W.     Food  Preservatives  .  . .  .8vo,  i  00 

paper,  o  50 

Eddy,  H.  T.     Researches  in  Graphical  Statics 8vo,  i  50 

Maximum  Stresses  under  Concentrated  Loads 8vo,  i  50 

Edgcumbe,  K.     Industrial  Electrical  Measuring  Instruments . 

8vo,  *2  50 

Eissler,  M.     The  Metallurgy  of  Gold 8vo,  7  50 

The  Hydrometallurgy  of  Copper 8vo,  *4  50 

The  Metallurgy  of  Silver 8vo,  4  00 

The  Metallurgy  of  Argentiferous  Lead 8vo,  5  00 

Cjranide  Process  for  the  Extraction  of  Gold 8vo,  3  00 

A  Handbook  of  Modern  Explosives 8vo,  5  00 

Ekin,  T.  C.     Water  Pipe  and  Sewage  Discharge  Diagrams,  .folio,  *3  00 
Eliot,  C.  W.,  and  Storer,  F.  H.    Compendious  Manual  of  Qualita- 
tive Chemical  Analysis i2mo,  *i  25 

Elliot,  Major  G.  H.     European  Light-house  Systems 8vo,  5  00 

Ennis,  Wm.  D.     Linseed  Oil  and  Other  Seed  Oils   8vo,  *4  00 

Applied  Thermodynamics 8vo,  *4  50 

Erfurt,  J.     Dyeing  of  Paper  Pulp.     Trans,  by  J.  Hubner. .  .8vo,  *7  50 

Erskine -Murray,  J.     A  Handbook  of  Wireless  Telegraphy.  .8vo,  *3  50 

Evans,  C.  A.     Macadamized  Roads {In  Press.) 

Ewing,  A.  J.     Magnetic  Induction  in  Iron 8vo,  *4  00 

Fairie,  J.     Notes  on  Lead  Ores i2mo,  *i  00 

Notes  on  Pottery  Clays i2mo,  *i  50 

Fairweather,  W.  C.     Foreign  and  Colonial  Patent  Laws  . .  .8vo,  *3  00 

Fanning,  T.  T.     Hydraulic  and  Water-supply  Engineering  .8vo,  *5  00 
Fauth,  P.     The  Moon  in  Modern  Astronomy.     Trans,  by  J. 

McCabe 8vo,  *2  00 

Fay,  I.  W.     The  Coal-tar  Colors 8vo  (In  Press.) 

Fernbach,  R.  L.    Glue  and  Gelatine. 8vo,  *3  00 


10     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Fischer,  E.     The  Preparation  of  Organic  Compounds.     Trans. 

by  R.  V.  Stanford 1 2mo,     *  i  25 

Fish,  J.  C.  L.     Lettering  of  Working  Drawings Oblong  8vo,       i  00 

Fisher,  H.  K.  C,  and  Darby,  W.  C.     Submarine  Cable  Testing. 

8vo,  *3  50 
Fiske,  Lieut.  B.  A.  Electricity  in  Theory  and  Practice  . .  .  .8vo,  2  50 
Fleischmann,  W.     The  Book  of  the  Dairy.     Trans,  by  C.  M. 

Aikman 8 vo,       4  00 

Fleming,    J.    A.     The    Alternate-current    Transformer.     Two 

Volumes 8vo, 

Vol.    I.     The  Induction  of  Electric  Currents *5  00 

Vol.  IL     The  Utilization  of  Induced  Currents *5  00 

Centenary  of  the  Electrical  Current 8vo,     *o  50 

Electric  Lamps  and  Electric  Lighting 8vo,     *3  00 

Electric  Laboratory  Notes  and  Forms 4to,     *5  00 

A  Handbook  for  the  Electrical  Laboratory  and  Testing 

Room.     Two  Volumes 8vo,  each,     *5  00 

Fluery,  H.     The  Calculus  Without  Limits  or  Infinitestimals. 

Trans,  by  C.  0.  Mailloux {In  Press) 

Foley,  N.     British  and  American  Customary  and  Metric  Meas- 
ures   folio,     *3  00 

Foster,     H.     A.     Electrical    Engineers'     Pocket-book.     {Sixth 

Edition.) i2mo,  leather,       5  00 

Foster,    Gen.    J.    G.     Submarine    Blasting    in    Boston    (Mass.) 

Harbor 4to,       3  50 

Fowle,  F.  F.     Overhead  Transmission  Line  Crossings  ..  .  .i2mo,     *i  50 

The  Solution  of  Alternating  Current  Problems. 

8vo  {In  Press) 
Fox,  W.,  and  Thomas,  C.  W.     Practical  Course  in  Mechanical 

Drawing 1 2mo,       i  25 

Francis,  J.  B.     Lowell  Hydraulic  Experiments 4to,     15  00 

Fuller,  G.  W.     Investigations  into  the  Purification  of  the  Ohio 

River 4to,  *io  00 

Furnell,  J.     Paints,  Colors,  Oils,  and  Varnishes 8vo,     *i  00 

Gant,  L.  W.     Elements  of  Electric  Traction 8vo,  ^2  50 

Garcke,  E.,  and  Fells,  J.  M.     Factory  Accounts 8vo,  3  00 

Garforth,  W.  E.     Rules  for  Recovering  Coal  Mines  after  Explo- 
sions and  Fires i2mo,  leather,  i  50 


I).  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     11 

Geerligs,  H.  C.  P.     Cane  Sugar  and  Its  Manufacture 8vo,  *5  oo 

Geikie,  J.     Structural  and  Field  Geology 8vo,  *4  oo 

Gerber,   N.     Analysis  of  Milk,   Condensed  Milk,  and  Infants' 

Milk-Food 8vo,  i  25 

Gerhard,  W.  P.     Sanitation,  Water-supply  and  Sewage  Disposal 

of  Country  Houses i2mo,  *2  00 

Gerhardi,  C.  W.  H.     Electricity  Meters Svo,  *4  00 

Geschwind,  L.     Manufacture  of  Alum  and  Sulphates.     Trans. 

by  C.  Salter Svo,  *5  00 

Gibbs,  W.  E.     Lighting  by  Acetylene i2mo,  *i  50 

Physics  of  Solids  and  Fluids.     (Carnegie  Technical  Schools 

Text-books.) *i  50 

Gibson,  A.  H.     Hydraulics  and  Its  Application Svo,  *$  00 

Water  Hammer  in  Hydraulic  Pipe  Lines i2mo,  *2  00 

Gillmore,  Gen.  Q.  A.  Limes,  Hydraulic  Cements  and  Mortars.Svo,  4  00 

Roads,  Streets,  and  Pavements i2mo,  2  00 

Golding,  H.  A.     The  Theta-Phi  Diagram i2mo,  *i  25 

Goldschmidt,  R.     Alternating  Current  Commutator  Motor  .Svo,  *3  00 

Goodchild,  W.     Precious  Stones.     (Westminster  Series.). .  .Svo,  *2  00 

Goodeve,  T.  M.     Textbook  on  the  Steam-engine i2mo,  2  00 

Gore,  G.     Electrolytic  Separation  of  Metals Svo,  *3  50 

Gould,  E.  S.     Arithmetic  of  the  Steam-engine i2mo,  i  00 

Practical  Hydrostatics  and  Hydrostatic  Formulas.     (Science 

Series.) i6mo,  o  50 

Grant,  J.    Brewing  and  Distilling.     (Westminster  Series.)  Svo  (/n  Press) 

Gray,  J.     Electrical  Influence  Machines i2mo,  2  00 

Greenwood,  E.     Classified  Guide  to  Technical  and  Commercial 

Books Svo,  *3  00 

Gregorius,  R.     Mineral  Waxes.     Trans,  by  C.  Salter i2mo,  *3  00 

Griffiths,  A.  B.     A  Treatise  on  Manures. i2mo,  3  00 

Dental  Metallurgy Svo,  *3  50 

Gross,  E.     Hops Svo,  *4  50 

Grossman,  J.     Ammonia  and  its  Compounds i2mo,  *i  25 

Groth,  L.  A.  Welding  and  Cutting  Metals  by  Gases  or  Electric- 
ity  Svo,  *3  00 

Grover,  F.     Modern  Gas  and  Oil  Engines Svo,  *2  00 

Gruner,  A.     Power-loom  Weaving Svo,  *3  00 

Guldner,    Hugo.      Internal-Combustion    Engines.      Trans,    by 

H.  Diedrichs 4to,  *io  00 


12     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Gunther,  C.  0.     Integration i2mo,     *i   25 

Gurden,  R.  L.     Traverse  Tables folio,  half  mor.       7  50 

Guy,  A.  E.     Experiments  on  the  Flexure  of  Beams 8vo,  *i  25 

Haeder,  H.     Handbook  on  the  Steam-engine.     Trans,  by  H.  H. 

P.  Powles i2mo, 

Hainbach,  R.     Pottery  Decoration.     Trans,  by  C.  Slater.    i2mo. 

Hale,  W.  J.     Calculations  of  General  Chemistry i2mo. 

Hall,  C.  H.     Chemistry  of  Paints  and  Paint  Vehicles. ....  i2mo. 

Hall,  R.  H.     Governors  and  Governing  Mechanism i2mo. 

Hall,  W.  S.     Elements  of  the  Differential  and  Integral  Calculus 

8vo, 

Descriptive  Geometry  . 8vo  volume  and  4to  atlas, 

Haller,  G.  F.,  and  Cunningham,  E.  T.    The  Tesla  Coil i2mo, 

Halsey,  F.  A.     SHde  Valve  Gears i2mo, 

The  Use  of  the  Slide  Rule.     (Science  Series.) i6mo, 

Worm  and  Spiral  Gearing.     (Science  Series.). ......  i6mo, 

Hamilton,  W.  G.     Useful  Information  for  Railway  Men. .  i6mo. 
Hammer, W.  J.     Radium  and  Other  Radioactive  Substances,  Svo, 

Hancock,  H.     Textbook  of  Mechanics  and  Hydrostatics Svo, 

Hardy,  E.     Elementary  Principles  of  Graphic  Statics i2mo. 

Harper,  W.  B.     Utilization  of  Wood  Waste  by  Distillation. .  4to, 

Harrison,  W.  B.     The  Mechanics'  Tool-book i2mo. 

Hart,  J.  W.     External  Plumbing  Work Svo, 

Hints  to  Plumbers  on  Joint  Wiping Svo, 

Principles  of  Hot  Water  Supply. Svo, 

Sanitary  Plumbing  and  Drainage. Svo, 

Haskins,  C.  H.     The  Galvanometer  and  Its  Uses i6mo, 

Hatt,  J.  A.  H.     The  Colorist. square  i2mo, 

Hausbrand,  E.     Drying  by  Means  of  Air  and  Steam.     Trans. 

by  A.  C.  Wright i2mo,     *2  00 

Evaporing,   Condensing   and   Cooling   Apparatus.     Trans. 

by  A.  C.  Wright. Svo,     *5  00 

Hausner,  A.     Manufacture  of  Preserved  Foods  and  Sweetmeats. 

Trans,  by  A.  Morris  and  H.  Robsoa.    Svo,     *3  00 

Hawke,  W.  H.     Premier  Cipher  Telegraphic  Code 4to,     *5  00 

100,000  Words  Supplement  to  the  Pre  nier  Code 4to,     *5  00 

Hawkesworth,  J.     Graphical  Handbook  for  Raniforced  Concrete 

Design 4to,     *2  50 


3 

00 

*3 

00 

*i 

00 

*2 

00 

*2 

00 

*2 

25 

*3 

50 

*i 

25 

I 

50 

0 

50 

0 

50 

I 

00 

*i 

00 

I 

50 

*i 

50 

*3 

GO 

I 

50 

*3 

00 

*3 

GO 

*3 

GO 

*3 

00 

I 

50 

*i 

50 

D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     13 

Hay,  A.  Alternating  Currents 8vo,  *2  50 

Principles  of  Alternate-current  "Working i2mo,  2  00 

Electrical  Distributing  Networks  and  Distributing  Lines.Svo,  *3  50 

Continuous  Current  Engineering 8vo,  *2  50 

Heap,  Major  D.  P.     Electrical  Appliances... 8vo,  2  00 

Heaviside,    0.     Electromagnetic    Theory.     Two    volumes. 

Svo,  each,  *5  00 
Heck,  R.  C.  H.     Steam-Engine  and  Other  Steam  Motors.     Two 

Volumes. 

Vol.    I.     Thermodynamics  and  the  Mechanics Svo,  *3  50 

Vol.  II.     Form,  Construction  and  Working Svo,  *5  00 

Abridged  edition  of  above  volumes  (Elementary) 

Svo  {In  Preparation) 

Notts  on  Elementary  Kinematics Svo,  boards,  *i  00 

Graphics  of  Machine  Forces Svo,  boards,  *i  00 

Hedges,  K.     Modern  Lightning  Conductors Svo,  3  00 

Heermann,  P.     Dyers' Materials.     Trnas.  by  A.  C.  Wright.i2mo,  *2  50 
Hellot,  Macquer  and  D'Apligny.     Art  of  Dyeing  Wool,  Silk  and 

Cotton Svo,  *2  00 

Henrici,  0.     Skeleton  Structures Svo,  i  50 

Hermann,  F.     Painting  on  Glass  and  Porcelain Svo,  *3  50 

Hermann,  G.     The  Graphical  Statics  of  Mechanism.     Trans. 

by  A.  P.  Smith i2mo,  2  00 

Herzfeld,  J.     Testing  of  Yarns  and  Textile  Fabrics Svo,  *3  50 

Hildebrandt,  A.     Airships,  Past  and  Present Svo,  *3  50 

Hill,  J.  W.     The  Purification  of  Public  Water  Supplies.     New 

Edition {In  Press.) 

Interpretation  of  Water  Analysis {In  Press.) 

Hiroi,  I.     Statically-Indeterminat'e  Stresses i2mo,  *2  00 

Hirshfeld,    C.    F.       Engineering     Thermodynamics.     (Science 

Series.) i6mo,  o  50 

Hobart,  H.  M.     Heavy  Electrical  Engineering.. Svo,  *4  50 

Electricity Svo,  *2  00 

Electric  Trains .Svo,  *2  50 

Hobbs,  W.  R.  P.     The  Arithmetic  of  Electrical  Measurements 

i2mo,  o  50 

Hoff,  J.  N.     Paint  and  Varnish  Facts  and  Formulas i2mo,  *3  00 

Hofif,  Com.W.  B.  The  Avoidance  of  Collisions  at  Sea.  i6mo,  mor.,  o  75 

Hole,  W.     The  Distribution  of  Gas Svo,  *7  50 


14     1).  VAN  NOSTRANI)  COMPANY'S  BHOKT-TITLE  CATALOG 

Holley,  A.  L.     Railway  Practice folio,  12  00 

Holmes,  A.  B.     The  Electric  Light  Popularly  Explained. 

i2mo,  paper,  o  50 

Hopkins,  N.  M.     Experimental  Electrochemistry... 8vo,  *3  00 

Model  Engines  and  Small  Boats i2mo,  i  25 

Horner,  J.     Engineers'  Turning 8vo,  *3  50 

Metal  Turning i2mo,  i  50 

Toothed  Gearing i2mo,  2  25 

Houghton,  C.  E.     The  Elements  of  Mechanics  of  Materials.  i2mo,  *2  00 

Houilevique,  L.     The  Evolution  of  the  Sciences 8vo,  *2  00 

Howe,  G.     Mathematics  for  the  Practical  Man i2mo  {In  Press.) 

Howorth,  J.     Repairing  and  Riveting  Glass,  China  and  Earthen- 
ware     Svo,  paper,  *o  50 

Hubbard,  E.     The  Utilization  of  Wood-waste Svo,  *2  50 

Humber,  W.     Calculation  of  Strains  in  Girders i2mo,  2  50 

Humphreys,    A.    C.     The    Business    Features    of    Engineering 

Practice Svo,  *  i  25 

Hurst,  G.  H.     Handbook  of  the  Theory  of  Color Svo,  *2  50 

Dictionary  of  Chemicals  and  Raw  Products Svo,  *3  00 

Lubricating  Oils,  Fats  and  Greases Svo,  *3  00 

Soaps Svo,  *5  00 

Textile  Soaps  and  Oils Svo,  *2  50 

Hutchinson,  R.  W.,  Jr.     Long  Distance  Electric  Power  Trans- 
mission  i2mo,  *3  00 

Hutchinson,   R.   W.,   Jr.,   and  Ihlseng,   M.   C.     Electricity  in 

Mining .  i2mo  {In  Press.) 

Hutchinson,  W.  B.     Patents  and  How  to  Make  Money  Out  of 

Them i2mo,  i  25 

Hutton,  W.  S.     Steam-boiler  Constructfon Svo,  6  00 

Practical  Engineer's  Handbook Svo,  7  00 

The  Works'  Manager's  Handbook Svo,  6  00 

Ingle,  H.     Manual  of  Agricultural  Chemistry Svo,  *3  00 

Innes,  C.  H.     Problems  in  Machine  Design i2mo,  *2  00 

Air  Compressors  and  Blowing  Engines .  i2mo,  *2  00 

Centrifugal  Pumps i2mo,  *2  00 

The  Fan i2mo,  *2  00 

Isherwood,  B.  F.     Engineering  Precedents  for  Steam  Machinery 

Svo,     2  50 


D.  VAN  NOSTIIAND  COMPANY'S  SHORT-TITLE  CATALOG     15 

Jamieson,  A.     Text  Book  on  Steam  and  Steam  Engines. .     8vo,  3  00 

Elementary  Manual  on  Steam  and  the  Steam  Engine.  1 2  mo,  i  50 

Jannettaz,  E.     Guide  to  the  Determination  of  Rocks.     Trans. 

by  G.  W.  Plympton i2mo,  i   50 

Jehl,  F.     Manufacture  of  Carbons 8vo,  *4  00 

Jennings,    A.   S.     Commercial   Paints   and   Painting.     (West- 
minster Series.) 8vo  (In  Press.) 

Jennison,  F.  H.     The  Manufacture  of  Lake  Pigments Svo,  *3  00 

Jepson,  G.     Cams  and  the  Principles  of  their  Construction...  Svo,  *i  50 

Mechanical  Drawing Svo  (In  Preparation.) 

Jockin,  W.     Arithmetic  of  the  Gold  and  Silversmith i2mo,  *i  00 

Johnson,  G.  L.     Photographic  Optics  and  Color  Photography 

Svo,  *3  00 
Johnson,    W.    H.     The  Cultivation  and   Preparation    of    Para 

Rubber Svo,  *3  00 

Johnson,  W.  McA.     The  Metallurgy  of  Nickel (In  Preparation.) 

Johnston,  J.  F.  W.,  and  Cameron,  C.     Elements  of  Agricultural 

Chemistry  and  Geology i2mo,  2  60 

Joly,  J.     Radioactivity  and  Geology i2mo,  *3  00 

Jones,  H.  C.     Electrical  Nature  of  Matter  and  Radioactivity 

i2mo,  2  00 

Jones,  M.  W.     Testing  Raw  Materials  Used  in  Paint. ...  i2mo,  *2  00 

Jones,  L.,  and  Scard,  F.  L     Manufacture  of  Cane  Sugar Svo,  *5  00 

Joynson,  F.  H.     Designing  and  Construction  of  Machine  Gear- 
ing  Svo,  2  00 

Jiiptner,  H.  F.  V.     Siderology:  The  Science  of  Iron Svo,  *5  00 

Kansas  City  Bridge 4to,  6  00 

Kapp,  G.     Electric  Transmission  of  Energy i2mo,  3  50 

Dynamos,    Motors,    Alternators    and    Rotary    Converters. 

Trans,  by  H.  H.  Simmons Svo,  4  00 

Keim,  A.  W.     Prevention  of  Dampness  in  Buildings  .....    Svo,  *2  00 
Keller,  S.  S.     Mathematics  for  Engineering  Students. 

i2mo,  half  leather, 

Algebra  and  Trigonometry,  with  a  Chapter  on  Vectors. ...  *i  75 

Special  Algebra  Edition *i  00 

Plane  and  Solid  Geometry. *i  25 

Analytical  Geometry  and  Calculus *2  00 

Kelsey,  W.  R.     Continuous-current  Dynamos  and  Motors..  Svo,  *2  50 


Kemble,  W.  T.,  and  Underbill,  C.  R.     The  Periodic  Law  and  the 

Hydrogen   Spectrum 8vo,  paper,  *o  50 

Kemp,  J.  F.     Handbook  of  Rocks 8vo,  *i  50 

Kendall,  E.     Twelve  Figure  Cipher  Code 4to,  *I5  00 

Kennedy,   R.     Modern  Engines  and   Power   Generators.     Six 

Volumes 4to,  15  00 

Single  Volumes each,  3  00 

Electrical  Installations.     Five  Volumes 4to,  15  00 

Single  Votumes each,  3  50 

Flying  Machines;  Practice  and  Design i2mo,  *2  00 

Kennelly,  A.  E.     Electro-dynamic  Machinery 8vo,  i  50 

Kershaw,  J.  B.  C.     Fuel,  Water  and  Gas  Analysis 8vo,  *2  50 

Electrometallurgy.     (Westminster  Series.) 8vo,  *2  00 

The  Electric  Furnace  in  Iron  and  Steel  Production.. i2mo,  *i  50 

Kingdon,  J.  A.     Applied  Magnetism 8vo,  *3  00 

Kinzbrunner,  C.     Alternate  Current  Windings 8vo,  *i  50 

Continuous  Current  Armatures 8vo,  *i  50 

Testing  of  Alternating  Current  Machines 8vo,  *2  00 

Kirkaldy,    W.    G.     David    Kirkaldy's    System    of    Mechanical 

Testing 4to,  10  00 

Kirkbride,  J.     Engraving  for  Illustration*. 8vo,  *i  50 

Kirkwood,  J.  P.     Filtration  of  River  Waters 4to,  7  50 

Klein,  J.  F.     Design  of  a  High  speed  Steam-engine 8vo,  *5  00 

Kleinhans,  F.  B.     Boiler  Construction 8vo,  3  00 

Knight,  Capt.  A.  M.     Modern  Steamship 8vo,  *7  50 

Half  Mor.  *9  00 

Knox,  W.  F.     Logarithm  Tables {In  Preparation.) 

Knott,  C.  G.,  and  Mackay,  J.  S.     Practical  Mathematics.  .  .8vo,  2  00 

Koester,  F.     Steam-Electric  Power  Plants 4to,  *5  00 

Hydroelectric  Developments  and  Engineering .4to,  *5  00 

Koller,  T.     The  Utilization  of  Waste  Products Bvo,  *3  50 

Cosmetics 8vo,  *2  50 

Krauch,  C.     Testing  of  Chemical  Reagents.     Trans,  by  J.  A. 

Williamson  and  L.  W.  Dupre 8vo,  *3  00 

Lambert,  T.     Lead  and  its  Compounds 8vo,  *3  50 

Bone  Products  and  Manures 8vo,  *3  00 

Lamborn,  L.  L.     Cottonseed  Products 8vo,  *3  00 

Jlodern  Soaps,  Candles,  and  Glycerin 8vo,  *7  50 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     17 

Lamprecht,  R.     Recovery  Work  After  Pit  Fires.      Trans,  by 

C.  Salter 8vo,  *4  oo 

Lanchester,  F.  W.     Aerial  Flight.     Two  Volumes.     8vo. 

Vol.    I.     Aerodynamics *6  oo 

Vol.  II.     Aerodonetics. *6  oo 

Larner,  E.  T.     Principles  of  Alternating  Currents i2mo,  *i  25 

Larrabee,   C.   S.     Cipher   and   Secret   Letter   and   Telegraphic 

Code i6mo,  o  60 

Lassar-Cohn,  Dr.     Modern  Scientific  Chemistry.     Trans,  by  M. 

M.  Pattison  Muir i2mo,  *2  00 

Latta,  M.  N.     Handbook  of  American  Gas-Engineering  Practice. 

8vo,  *4  50 

American  Producer  Gas  Practice 4to,  *6  00 

Leask,  A.  R.     Breakdowns  at  Sea. i2mo,  2  00 

Triple  and  Quadruple  Expansion  Engines i2mo,  2  00 

Refrigerating  Machinery i2mo,  2  00 

Lecky,  S.  T.  S.     "  "Wrinkles  "  in  Practical  Navigation Svo,  *8  00 

Leeds,  C.  C.    Mechanical  Drawing  for  Trade  Schools .  oblong,  4to, 

High  School  Edition *i  25 

Machinery  Trades  Edition *2  00 

Leflvre,  L.     Architectural  Pottery.     Trans,  by  H.  K.  Bird  and 

W.  M.  Binns 4to,  *7  50 

Lehner,  S.     Ink  Manufacture.     Trans,  by  A.  Morris  and  H. 

Robson 8vo,  *2  50 

Lemstrom,  S.     Electricity  in  Agriculture  and  Horticulture. 

8vo,  *i  50 
Lewes,  V.  B.     Liquid  and  Gaseous  Fuels.     (Westminster  Series.) 

SvOj  *2  00 

Lieber,  B.  F.     Lieber's  Standard  Telegraphic  Code Svo,  *io  00 

Code.     German  Edition Svo,  *io  00 

Spanish  Edition Svo,  *io  00 

French  Edition Svo,  *io  00 

Terminal  Index Svo,  *2  50 

Lieber's  Appendix folio,  *i5  00 

Handy  Tables 4to,  *2  50 

Bankers    and    Stockbrokers'    Code    and    Merchants    and 

Shippers'  Blank  Tables Svo,  *i5  00 

100,000,000  Combination  Code Svo,  *i5  00 

Engineering  Code Svo,  *io  00 


18     D.  VAN  NOSTRAND  COMPANY '8  SHOKT-TITLK  CATALOG 

Livermore,  V.  P.,  and  Williams,  J.     How  to  Become  a  Com- 
petent Motorman i2mo,  '''I  oo 

Livingst^e,  R.     Design  and  Construction  of  Commutators.  8vo,  *2  25 

Lobben,  P.     Machinists'  and  Draftsmen's  Handbook 8vo,  2  50 

Locke,  A.  G.  andC.  G.     Manufacture  of  Sulphuric  Acid 8vo,  10  00 

Lockwood,  T.  D.     Electricity,  Magnetism,  and  Electro-teleg- 

graphy Svo,  2  50 

Electrical  Measurement  and  the  Galvanometer i2mo,  i  50 

Lodge,  0.  J.     Elementary  Mechanics i2mo,  i  50 

Signalling  Across  Space  without  Wires.. Svo,  *2  00 

Lord,  R.  T.     Decorative  and  Fancy  Fabrics Svo,  *3  50 

Loring,  A.  E.     A  Handbook  of  the  Electromagnetic  Telegraph. 

i6mo,  o  50 
Lowenstein,  L.  C,  and  Crissey,  C.  P.     Centrifugal  Pumps.     (In  Press.) 

Lucke,  C.  E.     Gas  Engine  Design Svo,  *3  00 

Power  Plants:  their  Design,  Efficiency,  and  Power  Costs. 

2  vols (In  Preparation.) 

Power  Plant  Papers.     Form  I.     The  Steam  Power  Plant 

paper,  *i  50 

Lunge,  G.     Coal-tar  Ammonia.     Two  Volumes Svo,  *i5  00 

Manufacture  of  Sulphuric  Acid  and  Alkali.     Three  Volumes 

Svo, 

Vol.    I.     Sulphuric  Acid.     In  two  parts *I5  00 

Vol.  n.     Salt  Cake,  Hydrochloric  Acid  and  Leblanc  Soda. 

In  two  parts *i5  00 

VoL  III.     Ammonia  Soda *i5  00 

Technical  Chemists'  Handbook .  i2mo,  leather,  *3  90 

Technical  Methods  of  Chemical  Analysis.     Trans,  by  C.  A. 

Keane.     In  collaboration  with  the  corps  of  spacialists. 

Vol.    I.     In  two  parts Svo,  *i5  00 

Vols.  II  and  III (In  Preparation.) 

Lupton,  A.,  Parr,  G.  D.  A.,  and  Perkin,  H.     Electricity  as  Applied 

to  Mining Svo,  *4  50 

Luquer,  L.  M.     Minerals  in  Rock  Sections Svo,  *i  50 

Macewen,  H.  A.     Food  Inspection Svo,  *2  50 

Mackie,  J.     How  to  Make  a  Woolen  Mill  Pay Svo,  *2  00 

Mackrow,    C.     Naval    Architect's    and    Shipbuilder's    Pocket- 
book i6mo,  leather,  5  00 


*I 

50 

*I 

50 

*2 

00 

*I 

00 

3 

50 

*i 

00 

2 

50 

*2 

50 

*5 

00 

D.  VAN  NOSTKANI)  COMPANY'S  SHORT-TITLK  CATALOG     19 

Maguire,  Capt.  E.     The  Attack  and  Defense  of  Coast  Fortifica- 
tions  8vo,       2  50 

Maguire,  Wm.  R.     Domestic  Sanitary  Drainage  and  Plumbing 

8vo,       4  00 
Marks,  E.  C.  R.     Construction  of  Cranes  and  Lifting  Machinery 

i2mo, 

— —  Construction  and  Working  of  Pumps i2mo, 

Manufacture  of  Iron  and  Steel  Tubes i2mo, 

Mechanical  Engineering  Materials i2mo, 

Marks,  G.  C.     Hydraulic  Power  Engineering 8vo, 

— —  Inventions,  Patents  and  Designs. i2mo, 

Markham,  E.  R.     The  American  Steel  Worker i2mo, 

Marlow,  T.  G.     Drying  Machinery  and  Practice Svo, 

Marsh,  C.  F.     Concise  Treatise  on  Reinforced  Concrete.. .   Svo, 

Marsh,  C.  F.,  and  Dunn,  W.     Reinforced  Concrete 4to, 

Manual  of  Reinforced  Concrete  and  Concrete  Block  Con- 
struction  i6mo,  mor.,     *2  50 

Massie,  W.  W.,  and  Underbill,  C.  R.     Wireless  Telegraphy  and 

Telephony i2nio,     *i  00 

Matheson,  D.     Australian  Saw-Miller's  Log  and  Timber  Ready 

Reckoner i2nio,  leather,       i  50 

Mathot,  R.  E.     Internal  Combustion  Engines Svo, 

Maurice,  W.     Electric  Blasting  Apparatus  and  Explosives  .  .Svo, 

Shot  Firer's  Guide Svo, 

Maxwell,  W.  H.,  and  Brown,  J.  T.     Encyclopedia  of  Municipal 

and  Sanitary  Engineering 4to, 

Mayer,  A.  M.     Lecture  Notes  on  Physics.. Svo, 

McCullough,  R.  S.     Mechanical  Theory  of  Heat Svo, 

Mcintosh,  J.  G.     Technology  of  Sugar Svo, 

Industrial  Alcohol Svo,     *3  00 

Manufacture  of  Varnishes  and  Kindred  Industries.    Three 

Volumes.     Svo. 

VoL     I.     Oil  Crushing,  Refining  and  Boiling *3  50 

Vol.  II.     Varnish  Materials  and  Oil  Varnish  Making *4  00 

Vol.  III.. (In Preparation.) 

McMechen,  F.  L.     Tests  for  Ores,  Minerals  and  Metals..  .i2mo,     *i  00 

McNeill,  B.     McNeill's  Code Svo,     *6  00 

McPherson,  J.  A.     Water-works  Distribution Svo,       2  50 

Melick,  C.  W.     Dairy  Laboratory  Guide i2mo,     *i  25 


*3 

50 

*i 

50 

10 

00 

2 

00 

3 

50 

*4 

50 

*I 

50 

I 

50 

5 

00 

10 

oo 

*2 

50 

*I 

50 

*I 

oo 

*4 

00 

I 

50 

20     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Merck,  E.  Chemical  Reagents ;  Their  Purity  and  Tests ....  8vo, 
Merritt,  Wm.  H.  Field  Testing  for  Gold  and  Silver .  i6mo,  leather, 
Meyer,  J.  G.  A.,  and  Pecker,  C.  G.     Mechanical  Drawing  and 

Machine  Design 4to, 

Michell,  S.     Mine  Drainage 8vo, 

Mierzinski,  S.     Waterproofing  of  Fabrics.     Trans,  by  A.  Morris 

and  H.  Robson 8vo, 

Miller,  E.  H.     Quantitative  Analysis  for  Mining  Engineers .  .  8vo, 

Milroy,  M.  E.  W.     Home  Lace -making i2mo, 

Minifie,  W.     Mechanical  Drawing 8vo, 

Modern  Meteorology i2mo, 

Monckton,  C.  C.  F.     Radiotelegraphy.     (Westminster  Series.) 

8vo,      *2   00 
Monteverde,  R.  D.     Vest  Pocket  Glossary  of  English-Sf)anish, 

Spanish-English  Technical  Terms 64mo,  leather,     *i  oo 

Moore,  E.  C.  S.     New  Tables  for  the  Complete  Solution  of 

Ganguillet  and  Kutter's  Formula 8vo,     *5  oo 

Moreing,  C.  A.,  and  Neal,  T.     New  General  and  Mining  Tele- 
graph Code 8vo,     *5  oo 

Morgan,  A.  P.     Wireless  Telegraph  Construction  for  Amateurs. 

i2mo, 

Moses,  A.  J.     The  Characters  of  Crystals 8vo, 

Moses,  A.  J.,  and  Parsons,  C.  L  Elements  of  Mineralogy.  .8vo, 
Moss,    S.    A.     Elements    of    Gas    Engine    Design.     (Science 

Series.) i6mo, 

The  Lay-out  of  Corliss  Valve  Gears.     (Science  Series) .  i6mo, 

MuUin,  j.  P.  Modern  Moulding  and  Pattern- making .  .  .  .  i2mo, 
Munby,  A.  E.     Chemistry  and  Physics  of  Building  Materials. 

(Westminster  Series.). 8vo, 

Murphy,  J.  G.     Practical  Mining i6mo, 

Murray,  J.  A.     Soils  and  Manures.     (Westminster  Series.). 8 vo, 

Naquet,  A.     Legal  Chemistry i2mo, 

Nasmith,  J.     The  Student's  Cotton  Spinning 8vo, 

Nerz,  F.     Searchlights.     Trans,  by  C.  Rodgers 8vo, 

Neuberger,   H.,   and   Noalhat,   H.     Technology   of   Petroleum. 

Trans,  by  J.  G.  Mcintosh. 8vo,  *io  oo 

Newall.  J.  W.      Drawing,  Sizing  and   Cutting  Bevel-gears. 

8vo,       I  50 


*I 

50 

*2 

00 

*2 

50 

0 

50 

O 

50 

2 

50 

*2 

00 

I 

00 

*2 

00 

2 

00 

3 

00 

*3 

00 

15 

00 

*4 

50 

I 

00 

*3 

00 

I 

50 

I 

50 

3 

50 

*o 

75 

*4 

00 

*o 

50 

*3*t)0 

21 


Newlands,  J.     Carpenters  and  Joiners'  Assistant 

folio,  half  mor., 

Nicol,  G.     Ship  Construction  and  Calculations 8vo, 

Nipher,  F.  E.     Theory  of  Magnetic  Measurements. i2mo, 

Nisbet,  H.     Grammar  of  Textile  Design 8vo, 

Noll,  A.     How  to  Wire  Buildings i2mo, 

Nugent,  E.     Treatise  on  Optics i2mo, 

O'Connor,  H.     The  Gas  Engineer's  Pocketbook. .  .  i2mo,  leather, 

Petrol  Air  Gas i2mo, 

Olsen,  J.  C.     Text  book  of  Quantitative  Chemical  Analysis  .  .8vo, 

Olsson,  A.     Motor  Control,  in  Turret  Turning  and  Gun  Elevating. 

(U.  S.  Navy  Electrical  Series,  No.  i.) .  ...i2mo,  paper, 

Oudin,  M.  A.     Standard  Polyphase  Apparatus  and  Systems  . ,  8  vo, 

Palaz,  A.     Industrial  "Photometry.     Trans,  by  G.  W.  Patterson, 

Jr 8vo,  *4  00 

Pamely,  C.     Colliery  Manager's  Handbook 8vo,  *io  00 

Parr,  G.  D.  A.     Electrical  Engineering  Measuring  Instruments. 

8vo,  *3  50 
Parry,  E.  J.     Chemistry  of  Essential  Oils  and  Artificial  Per- 
fumes  8vo,  *5  00 

Parry,  E.  J.,  and  Coste,  J.  H.     Chemistry  of  Pigments 8vo,  *4  50 

Parry,  L.  A.     Risk  and  Dangers  of  Various  Occupations 8vo,  *3  00 

Parshall,  H.  F.,  and  Hobart,  H.  M.     Armature  Windings  ....  4to,  *7  50 

Electric  Railway  Engineering. 4to,  *io  00 

Parshall,  H.  F.,  and  Parry,  E.     Electrical  Equipment  of  Tram- 
ways  (In.Press.) 

Parsons,  S.  J.     Malleable  Cast  Iron 8vo,  *2  50 

Passmore,  A.  C.     Technical  Terms  Used  in  Architecture  ...8vo,  *3  50 

Patterson,  D.     The  Color  Printing  of  Carpet  Yarns 8vo,  *3  50 

Color  Matching  on  Textiles 8vo,  *3  00 

X The  Science  of  Color  Mixing 8vo,  *3  00 

Patton,  H.  B.     Lecture  Notes  on  Crystallography 8vo,  *i  25 

Paulding,  C.  P.     Condensation  of  Steam  in  Covered  and  Bare 

Pipes. 8vo,  *2  00 

Transmission  of  Heat  through  Cold-storage  Insulation 

i2mo,  *i  00 

Peirce,  B.     System  of  Analytic  Mechanics 4to,  10  00 


22   D. 

Pendred,  V.     The  Railway  Locomotive.     (Westminster  Series.) 

8V0,  *2    00 

Perkin,  F.  M.  Practical  Methods  of  Inorganic  Chemistry.  i2mo,  *i  oo 

Perrigo,  0.  E.     Change  Gear  Devices 8vo,  i  oo 

Perrine,  F.  A.  C.     Conductors  for  Electrical  Distribution  .  .  .  8vo,  *3  50 

Perry,  J.     Applied  Mechanics 8vo,  *2  50 

Petit,  G.     White  Lead  and  Zinc  White  Paints 8vo,  *i  50 

Petit,   R.     How  to  Build  an  Aeroplane.     Trans,   by  T.   O'B. 

Hubbard,  and  J.  H.  Ledeboer 8vo,  *i  50 

Phillips,  J.     Engineering  Chemistry 8vo,  *4  50 

Gold  Assaying 8vo,  *2  50 

Phin,  J.     Seven  Follies  of  Science i2mo,  *i  25 

Household  Pests,  and  How  to  Get  Rid  of  Them 

8vo  {In  Preparation.) 
Pickworth,  C.  N.     The  Indicator  Handbook.     Two  Volumes 

i2mo,  each,  i  50 

Logarithms  for  Beginners i2mo,  boards,  o  50 

The  Slide  Rule i2mo,  i  00 

Plane  Table,  The 8vo,  2  00 

Plattner's  Manual  of    Blowpipe  Analysis.     Eighth  Edition,  re- 
vised.    Trans  by  H.  B.  Cornwall 8vo,  *4  00 

Plympton,  G.  W.     The  Aneroid  Barometer.     (Science  Series.) 

i6mo,  o  50 

Pocket  Logarithms  to  Four  Places.     (Science  Series.). ....  i6mo,  o  50 

leather,  i  00 

Pope,  F.  L.     Modern  Practice  of  the  Electric  Telegraph. . .   8vo,  i  50 
Popplewell,  W.  C.     Elementary  Treatise   on  Heat  and  Heat 

Engines i2mo,  *3  00 

Prevention  of  Smoke 8vo,  *3  50 

Strength  of  Minerals 8vo,  *  i  75 

Potter,  T.     Concrete 8vo,  *3  00 

Practical  Compounding  of  Oils,  Tallow  and  Grease 8vo,  *3  50 

Practical  Iron  Founding i2mo,  i  50 

Pray,  T.,  Jr.     Twenty  Years  with  the  Indicator 8vo,  2  50 

Steam  Tables  and  Engine  Constant 8vo,  2  00 

Calorimeter  Tables 8vo,  i  00 

Preece,  W.  H.     Electric  Lamps {In  Press.) 

Prelini,  C.     Earth  and  Rock  Excavation 8vo,  *3  00 

Graphical  Determination  of  Earth  Slopes. 8vo,  *2  00 


D.  VAN  NOSTRAND  COMPANY^S  SHORT  TITLE  CATALOG     23 

Prelini,  C.     Tunneling 8vo,  3  00 

Dredges  and  Dredging 8vo  {In  Press.) 

Prescott,  A.  B.     Organic  Analysis. 8vo,  5  00 

Prescott,   A.   B.,   and  Johnson,   0.   C.     Quauitative   Chemical 

Analysis Svo,  *3  50 

Prescott,  A.  B.,  and  Sullivan,  E.  C.     First  Book  in  Qualitative 

Chemistry i2mo,  *i  50 

Pritchard,  0.  G.     The  Manufacture  of  Electric-light  Carbons. 

Svo,  paper,  *o  60 
Prost,  E.     Chemical  Analysis  of  Fuels,  Ores,  Metals.     Trans. 

by  J.  C.  Smith Svo,  *4  50 

Pullen,  W.  W.  F.     Application  of  Graphic  Methods  to  the  Design 

of  Structures i2mo,  *2  50 

Injectors:  Theory,  Construction  and  Working..  ....  i2mo,  *i  50 

Pulsifer,  W.  H.     Notes  for  a  History  of  Lead. Svo,  4  00 

Putsch,  A.     Gas  and  Coal-dust  Firing Svo,  *3  00 

Pynchon,  T.  R.     Introduction  to  Chemical  Physics Svo,  3  00 

Rafter,  G.  W.     Treatment  of  Septic  Sewage.     (Science  Series.) 

i6mo,  o  50 
Rafter,  G.  W.,  and  Baker,  M.  N.     Sewage  Disposal  in  the  United 

States 4to,  *6  00 

Raikes,  H.  P.     Sewage  Disposal  Works Svo,  *4  00 

Railway  Shop  Up-to-Date 4to,  2  00 

Ramp,  H.  M.     Foundry  Practice {In  Press.) 

Randall,  P.  M.     Quartz  Operator's  Handbook i2mo,  2  00 

Randau,  P.     Enamels  and  Enamelling Svo,  *4  00 

Rankine,  W.  J.  M.     Applied  Mechanics Svo,  5  00 

Civil  Engineering Svo,  6  50 

Machinery  and  Millwork. Svo,  5  00 

The  Steam-engine  and  Other  Prime  Movers Svo,  5  00 

Useful  Rules  and  Tables .    Svo,  4  00 

Rankine,  W.  J.  M.,  and  Bamber,  E.  F.     A  Mechanical  Text- 
book  Svo,  3  50 

Raphael,  F.  C.     Localization  of    Faults  in  Electric  Light  and 

Power  Mains Svo,  *3  00 

Rathbone,  R.  L.  B.     Simple  Jewellery Svo,  *2  00 

Rateau,   A.     Flow  of  Steam  through   Nozzles    and    Orifices. 

Trans,  by  H.  B.  Brydon ..Svo,  *i  50 


24     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Rausenberger,  F.     The  Theory  of  the  Recoil  of  Guns. ....  8vo,     *4  50 
Rautenstrauch,  W.     Notes  on  the  Elements  of  Machine  Design, 

8vo,  boards,     *i  50 
Rautenstrauch,  W.,  and  Williams,  J.  T.     Machine  Drafting  and 

Empirical  Design. 

Part   I.  Machine  Drafting 8vo,     *i  25 

Part  II.  Empirical  Design (In  Preparatio7i.) 

Raymond,  E.  B.     Alternating  Current  Engineering i2mo, 

Rayner,  H.     Silk  Throwing  and  Waste  Silk  Spinning Svo, 

Recipes  for  the  Color,  Paint,  Varnish,  Oil,  Soap  and  Drysaltery 

Trades Svo, 

Recipes  for  Flint  Glass  Making i2mo. 

Reed's  Engineers'  Handbook Svo, 

Key  to  the  Nineteenth  Edition  of  Reed's  Engineers'  Hand- 
book  Svo, 

Useful  Hints  to  Sea-going  Engineers i2mo, 

Marine  Boilers. i2mo, 

Reinhardt,   C.  W.     Lettering  for  Draftsmen,   Engineers,  and 

Students oblong  4to,  boards, 

The  Technic  of  Mechanical  Drafting. . .  oblong  4to,  boards, 

Reiser,  F.     Hardening  and  Tempering  of  Steel.     Trans,  by  A. 

Morris  and  H.  Robson i2mo,     *2  50 

Reiser,  N.     Faults  in  the  Manufacture  of  Woolen  Goods.     Trans. 

by  A.  Morris  and  H.  Robson Svo, 

Spinning  and  Weaving  Calculations Svo, 

Renwick,  W.  G.     Marble  and  Marble  Working Svo, 

Rhead,  G.  F.     Simple  Structural  Woodwork i2mo, 

Rhead,  G.  W.     British  Pottery  Marks Svo, 

Rice,  J.  M.,  and  Johnson,  W.  W.     A  New  Method  of  Obtaining 

the  Differential  of  Functions i2mo, 

Richardson,  J.     The  Modern  Steam  Engine Svo, 

Richardson,  S.  S.     Magnetism  and  Electricity i2mo, 

Rideal,  S.     Glue  and  Glue  Testing Svo, 

Rings,  F.     Concrete  in  Theory  and  Practice i2mo. 

Ripper,  W.     Course  of  Instruction  in  Machine  Drawing. .   folio, 
Roberts,  J.,  Jr.      Laboratory  Work  in  Electrical  Engineering. 

Svo, 

Robertson,  L.  S.     Water-tube  Boilers Svo, 

Robinson,  J.  B.     Architectural  Composition Svo, 


*2 

50 

*2 

50 

*3 

50 

*4 

50 

*5 

00 

*3 

00 

I 

50 

2 

00 

I 

00 

*i 

00 

*2 

50 

*5 

00 

5 

00 

*i 

00 

*3 

00 

0 

50 

*3 

50 

*2 

00 

*4 

00 

*2 

50 

*6 

00 

*2 

00 

3 

00 

*2 

50 

D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG  25 

Robinson,  S.  W.     Practical  Treatise  on  the  Teeth  of  Wheels. 

(Science  Series.) i6mo,  o  50 

Roebling,  J.  A.     Long  and  Short  Span  Railway  Bridges.  .   folio,  25  00 

Rogers,  A.     A  Laboratory  Guide  of  Industrial  Chemistry.  .  i2mo,  *i  50 

Rogers,  A.,  and  Aubert,  A.  B.     Industrial  Chemistry {In  Press.) 

Rollins,  W.     Notes  on  X-Light 8vo,  *7  50 

Rose,  J.     The  Pattern-makers'  Assistant 8vo,  2  50 

• Key  to  Engines  and  Engine-running i2mo,  2  50 

Rose,  T.  K.      The  Precious  Metals.      (Westminster  Series.). 

8vo,  *2  00 

Rosenhain,  W.  Glass  Manufacture.  (Westminster  Series.) .  .Svo,  *2  00 
Rossiter,  J.  T.     Steam  Engines.     (Westminster  Series.) 

Svo  (In  Press.) 
Pumps >  and  Pumping  Machinery.     (Westminster  Series.) 

Svo  (In  Press.) 

Roth.     Physical  Chemistry Svo,  *2  00 

Rouillion,  L.     The  Economics  of  Manual  Training Svo,  2  00 

Rowan,  F.  J.     Practical  Physics  of  the  Modern  Steam-boiler 

Svo,  7  50 

Roxburgh,  W.     General  Foundry  Practice Svo,  *3  50 

Ruhmer,    E.     Wireless    Telephony.     Trans,    by    J.    Erskine- 

Murray Svo,  *3  50 

Russell,  A.     Theory  of  Electric  Cables  and  Networks Svo,  *3  00 

Sabine,  R.  History  and  Progress  of  the  Electric  Telegraph.  i2mo,  i  25 

Saeltzer,  A.     Treatise  on  Acoustics i2mo,  i  00 

Salomons,  D.     Electric  Light  Installations.     i2mo. 

Vol.     I.     The  Management  of  Accumulators 2  50 

Vol.    II.     Apparatus 2  25 

Vol.  III.     Applications i  50 

Sanford,  P.  G.     Nitro-explosives Svo,  *4  00 

Saunders,  C.  H.     Handbook  of  Practical  Mechanics i6mo,  i  00 

leather,  i  25 

Saunnier,  C.     Watchmaker's  Handbook i2mo,  3  00 

Sayers,  H.  M.     Brakes  for  Tram  Cars Svo,  *i  25 

Scheele,  C.  W.     Chemical  Essays Svo,  *2  00 

Schellen,  H.     Magneto-electric  and  Dynamo -electric  Machines 

Svo,  5  00 

Scherer,  R.     Casein.     Trans,  by  C.  Salter. Svo,  *3  po 


26     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Schmall,  C.  N.     First  Course  in  Analytic  Geometry,  Plane  and 

Solid. i2mo,  half  leather,  *i  75 

Schmall,  C.  N.,  and  Schack,  S.  M.     Elements  of  Plane  Geometry 

i2mo,  *i  25 

Schmeer,  L,     Flow  of  Water 8vo,  *3  00 

Schumann,  F.     A  Manual  of  Heating  and  Ventilation. 

i2mo,  leather,  i  50 

Schwartz,  E.  H.  L.     Causal  Geology 8vo,  *2  50 

Schweizer,  V.,  Distillation  of  Resins 8vo,  *3  50 

Scott,  W,  W.     Qualitative  Chemical  Analysis.     A  Laboratory 

Manual Bvo  {In  Press.) 

Scribner,  J.  M.     Engineers'  and  Mechanics'  Companion. 

7                                                             i6mo,  leather,  i  50 
Searle,  G.  M.     "  Sumners'  Method."     Condensed  and  Improved. 

(Science  Series.) i6mo,  o  50 

Seaton,  A.  E.     Manual  of  Marine  Engineering. Bvo,  6  00 

Seaton,  A.  E.,  and  Rounthwaite,  H.  M.     Pocket-book  of  Marine 

Engineering i6mo,  leather,  3  00 

Seeligmann,   T.,   Torrilhon,   G.   L.,   and  Falconnet,   H.     India 

Rubber  and  Gutta  Percha.     Trans,  by  J.  G.  Mcintosh 

8vo,  *5  00 
Seidell,  A.     Solubilities  of  Inorganic  and  Organic  Substances 

8vo,  New  Edition  (In  Preparation.) 

Sellew,  W.  H.     Steel  Rails 4to  (/n  Press.) 

Senter,  G.     Outlines  of  Physical  Chemistry i2mo,  *i  50 

Sever,  G.  F.     Electric  Engineering  Experiments  ....  8vo,  boards,  *i  00 
Sever,  G.  F.,  and  Townsend,  F.     Laboratory  and  Factory  Tests 

in  Electrical  Engineering 8vo,  *2  50 

Sewall,  C.  H.     Wireless  Telegraphy 8vo,  *2  00 

Lessons  in  Telegraphy i2mo,  *i  00 

Sewell,  T.     Elements  of  Electrical  Engineering 8vo,  *3  00 

The  Construction  of  Dynamos Bvo,  *3  00 

Sexton,  A.  H.     Fuel  and  Refractory  Materials i2mo,  *2  50 

Chemistry  of  the  Materials  of  Engineering .  . i2mo,  *2  50 

Alloys  (Non- Ferrous) 8vo,  *3  00 

The  Metallurgy  of  Iron  and  Steel 8vo,  *6  50 

Seymour,  A.     Practical  Lithography Bvo,  *2  50 

Modern  Printing  Inks Bvo,  *2  00 

Shaw,  P.  E,     Course  of  Practical  Magnetism  and  Electricity.  8 vo,  *  i  00 


27 


Shaw,  S.     History  of  the  Staffordshire  Potteries 8vo,  *3  oo 

Chemistry  of  Compounds  Used  in  Porcelain  Manufacture 

8vo,  *5  oo 
Sheldon,  S.,  and  Hausmann,  E.      Direct   Current  Machines. 

8vo,  *2  50 
Sheldon,  S.,  Mason,  H.,  and  Hausmann,  E.     Alternating-curreat 

Machines 8vo,  *2  50 

Sherer,  R.     Casein.     Trans,  by  C.  Salter *.  .  .  .  8vo,  *3  00 

Sherriff,  F.  F.     Oil  Merchants'  Manual i2mo,  *3  50 

Shields,  J.  E.     Notes  on  Engineering  Construction.. i2mo,  i  50 

Shock,  W.  H.     Steam  Boilers. 4to,  half  mor.,  15  00 

Shreve,  S.  H.     Strength  of  Bridges  and  Roofs  .  - 8vo,  3  50 

Shunk,  W.  F.     The  Field  Engineer. i2mo,  mor.,  2  50 

Simmons,  W.   H.,  and  Appleton,   H.   A.     Handbook  of  Soap 

Manufacture 8vo,  *3  00 

Simms,  F.  W.     The  Principles  and  Practice  of  Leveling 8vo,  2  50 

Practical  Tunneling 8vo,  7  50 

Simpson,  G.     The  Naval  Constructor i2mo,  mor.,  *5  00 

Sinclair,  A.     Development  of  the  Locomotive  Engine. 

8vo,  half  leather,  5  00 
Sindall,  R.  W.     Manufacture  of  Paper.     (Westminster  Series.) 

8vo,  *2  00 

Sloane,  T.  O'C.     Elementary  Electrical  Calculations  ....  i2mo,  *2  00 

Smith,  C.  F.     Practical  Alternating  Currents  and  Testing .  .  8vo,  *2  50 

Practical  Testing  of  Dvnamos  and  Motors 8vo,  *2  00 

Smith,  F.  E.     Handbook  of  General  Instruction  for  Mechanics. 

i2mo,  I  50 
Smith,  I.  W.     The  Theory  of  Deflections  and  of  Latitudes  and 

Departures i6mo,  mor.,  3  00 

Smith,  J.  C.     Manufacture  of  Paint 8vo,  *3  00 

Smith,  W.     Chemistry  of  Hat  Manufacturing i2mo,  *3  00 

Snell,  A.  T.     Electri'c  Motive  Power. 8vo,  *4  00 

Snow,  W.  G.     Pocketbook  of  Steam  Heating  and  Ventilation 

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Series.) i6mo,  o  50 

Soddy,  F.     Radioactivity 8vo,  *3  00 

Solomon,  M.     Electric  Lamps.     (Westminster  Series.) 8vo,  *2  00 

Sothern,  J.  W.     The  Marine  Steam  Turbine 8vo.  *5  00 


28     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Soxhlet,  D.  H.     Dyeing  and  Staining  Marble.     Trans,  by  A. 

Morris  and  H.  Robson 8vo,     *2  50 

Spang,  H.  W.     A  Practical  Treatise  on  Lightning  Protection 

i2mo, 

Speyers,  C.  L.     Text-book  of  Physical  Chemistry 8vo, 

Stahl,  A,  W.,  and  Woods,  A.  T.     Elementary  Mechanism .  .  1 2mo, 
Staley,  C,  and  Pierson,  G.  S.     The  Separate  System  of  Sewerage. 

8vo, 

Standage,  H.  C.     Leatherworkers'  Manual Bvo, 

Sealing  Waxes,  Wafers,  and  Other  Adhesives Bvo, 

Agglutinants  of  all  Kinds  for  all  Purposes i2mo, 

Stansbie,  J.  H.     Iron  and  SteeL     (Westminster  Series.) ....  8vo, 

Stevens,  H.  P.     Paper  Mill  Chemist i6mo, 

Stewart,  A.     Modern  Polyphase  Machinery i2mo, 

Stewart,  G.     Modern  Steam  Traps i2mo. 

Stiles,  A.     Tables  for  Field  Engineers i2mo, 

Stillman,  P.     Steam-engine  Indicator i2mo, 

Stodola,  A.     Steam  Turbines.     Trans,  by  L.  C.  Loewenstein .  Bvo, 

Stone,  H.     The  Timbers  of  Commerce Bvo, 

Stone,  Gen.  R.     New  Roads  and  Road  Laws i2mo, 

Stopes,  M.     Ancient  Plants Bvo, 

Sudborough,  J.  J.,  and  James,  T.  C.     Practical  Organic  Chem- 
istry  i2mo, 

Suflling,  E.  R.     Treatise  on  the  Art  of  Glass  Painting Bvo, 

Swan,  K.     Patents,  Designs  and  Trade  Marks.     (Westminster 

Series.) Bvo, 

Sweet,  S.  H.     Special  Report  on  Coal Bvo, 

Swoope,  C.  W.     Practical  Lessons  in  Electricity i2mo, 

Tailfer,  L.     Bleaching  Linen  and  Cotton  Yam  and  Fabrics     Bvo,  *5  00 
Templeton,  W.     Practical  Mechanic's  Workshop  Companion. 

i2mo,  mor.,  2  00 
Terry,  H.  L.     India  Rubber  and  its  Manufacture.     (Westminster 

Series.) Bvo,  *2  00 

Thom,  C,  and  Jones,  W.  H.     Telegraphic  Connections. 

oblong  i2mo,  i  50 

Thomas,  C.  W.     Paper-makers'  Handbook (In  Press.) 

Thompson,  A.  B.     Oil  Fields  of  Russia 4to,  *7  50 

Petroleum  Mining  and  Oil  Field  Development Bvo,  *5  00 


I 

00 

*2 

25 

*2 

00 

*3 

00 

*3 

50 

*2 

00 

*3 

50 

*2 

00 

*2 

50 

*2 

00 

*I 

25 

I 

00 

I 

00 

*5 

00 

3 

50 

I 

00 

*2 

00 

*2 

00 

*3 

50 

*2 

00 

3 

00 

*2 

00 

I 

50 

*3 

00 

*i 

50 

*2 

50 

*2 

50 

*4 

00 

*3 

00 

*2 

00 

*3 

00 

*7 

50 

*2 

00 

*I 

75 

D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLR  CATALOG     29 

Thompson,  E.  P.     How  to  Make  Inventions 8vo,       o  50 

Thompson,  W.  P.     Handbook  of  Patent  Law  of  All  Countries 

i6mo, 

Thornley,  T.     Cotton  Combing  Machines 8vo, 

Cotton  Spinning 8vo, 

First  Year 

Second  Year 

Third  Year. 

Thurso,  J.  W.     Modern  Turbine  Practice Svo, 

Tinney,  W.  H.     Gold-mining  Machinery Svo, 

Titherley,  A.  W.     Laboratory  Course  of  Organic  Chemistry. .  Svo, 

Toch,  M.     Chemistry  and  Technology  of  Mixed  Paints Svo, 

Todd,  J.,  and  Whall,  W.  B.     Practical  Seamanship Svo, 

Tonge,  J.     Coal.     (Westminster  Series.) Svo, 

Townsend,  J.     Ionization  of  Gases  by  Collision Svo, 

Transactions  of  the  American  Institute  of  Chemical  Engineers. 

Svo, 

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VoL  II.     1909 *6  00 

Traverse  Tables.     (Science  Series.) i6mo, 

mor., 
Trinks,    W.,    and    Housum,    C.      Shaft    Governors.     (Science 

Series.) i6mo, 

Tucker,  J.  H.     A  Manual  of  Sugar  Analysis Svo, 

Tumlirz,  0.     PotentiaL     Trans,  by  D.  Robertson 12 mo, 

Tunner,   P.   A.     Treatise   on   Roll-turning.     Trans,   by  J.   B. 

Pearse Svo  text  and  folio  atlas, 

Turbayne,  A.  A.     Alphabets  and  Numerals 4to, 

Turrill,  S.  M.     Elementary  Course  in  Perspective 12 mo. 

Underbill,  C.  R.     Solenoids,  Electromagnets  and  Electromag- 
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Urquhart,  J.  W.     Electric  Light  Fitting i2mo, 

Electro-plating i2mo, 

Electrotyping i2mo, 

Electric  Ship  Lighting i2mo, 

Universal  Telegraph  Cipher  Code \  .  i2mo, 

Vacher,  F.     Food  Inspector's  Handbook i2mo,     *2  50 


0 

50 

I 

00 

0 

50 

3 

50 

I 

25 

10 

00 

2 

00 

*i 

25 

*2 

00 

2 

00 

2 

00 

2 

00 

3 

00 

I 

00 

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Year  Book  of  Mechanical  Engineering  Data.     First  issue 

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Van  Wagenen,  T.  F.     Manual  cf  Hydraulic  Mining. i6mo,  i  00 

Vega,   Baron,   Von     Logarithmic  Tables 8vo,  half  mor.,  2  50 

Villon,    A.    M.     Practical    Treatise    on   the    Leather   Industry. 

Trans,  by  F.  T.  Addyman 8vo,  *io  00 

Vincent,  C.     Ammonia  and  its  Compounds.     Trans,  by  M.  J. 

Salter 8vo,  *2  00 

Volk,  C.     Haulage  and  Winding  Appliances Svo,  *4  00 

Von  Georgiovics,  G.     Chemical  Technology  of  Textile  Fibres. 

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Chemistry  of  Dyestuffs.     Trans,  by  C.  Salter Svo,  *4  50 

Wabner,  R.     Ventilation  in  Mines.     Trans,  by  C.  Salter.  .   Svo,  *4  50 

Wade,  E.  J.     Secondary  Batteries Svo,  *4  00 

Wadsworth,  C.     Primary  Battery  Ignition i2mo  (In  Press.) 

Wagner,  E.     Preserving  Fruits,  Vegetables,  and  Meat i2mo,  *2  50 

Walker,  F.     Aerial  Navigation Svo, 

Electric  Lighting  for  Marine  Engineers Svo,  2  00 

Walker,  S.  F.     Steam  Boilers,  Engines  and  Turbines Svo,  3  00 

Refrigeration,  Heating  and  Ventilation  on  Shipboard. 

i2mo,  *2  00 

Electricity  in  Mining Svo,  *3  50 

Walker,  W.  H.     Screw  Propulsion Svo,  o  75 

Wallis-Tayler,  A.  J.     Bearings  and  Lubrication Svo,  *i  50 

Modern  Cycles Svo,  4  00 

Motor  Cars Svo,  i  So 

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Sugar  Machinery. i2mo,  *2  00 

Wanklyn,  J.  A.     Treatise  on  the  Examination  of  Milk     .  .  i2mo,  i  00 

Water  Analysis i2mo,  2  00 

Wansbrough,  W.  D.     The  A.  B  C  of  the  Differential  Calculus 

i2mo,  *i  50 

Slide  Valves i2mo,  *2  00 

Ward,  J.  H.     Steam  for  the  Million. Svo,  i  00 


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Waring,  G.  E.,  Jr.     Sewerage  and  Land  Drainage *6  oo 

Modern  Methods  of  Sewage  Disposal i2mo,  2  00 

How  to  Drain  a  House. i2mo,  i  25 

Warren,  F.  D.     Handbook  on  Reinforced  Concrete i2mo,  *2  50 

Watkins,  A.     Photography.     (Westminster  Series) 8vo  (In  Press.) 

Watson,  E.  P.     Small  Engines  and  Boilers i2mo,  i  25 

Watt,  A.     Electro-plating  and  Electro-refining  of  Metals *4  5© 

Watt,  A.     Electro-metallurgy i2mo,  i  00 

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Weale,  J.     Dictionary  of  Terms  used  in  Architecture i2mo,  2  50 

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paper,  o  50 
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Webber,  W.  H.  Y.     Town  Gas.     (Westminster  Series.) Svo,  *2  00 

Weekes,  R.  W.     The  Design  of  Alternate  Current  Transformers 

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sheep,  *7  50 
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Whipple,  S.     An  Elementary  and  Practical  Treatise  on  Bridge 

Building Svo,  3  00 

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Williams,  A.  D.,  Jy.,  and  Hutchinson,  R.  W.    The  Steam  Turbine. 

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Williamson,  R.  S.     On  the  Use  of  the  Barometer 4to,  15  00 

Practical  Tables  in  Meteorology  and  Hypsometery 4to,  2  50 

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Winkler,  C,  and  Lunge,  G.     Handbook  of  Technical  Gas-Analy- 
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Woodbury,  D.  V.     Elements  of  Stability  in  the  Well-propor- 
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Worden,  E.  C.     The  Nitrocellulose  Industry.     Two  Volumes. 

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Simple  Method  for  Testing  Painter's  Materials 8vo,  *2  50 

Wright,  H.  E.     Handy  Book  for  Brewers Bvo,  *5  00 

Wright,  F.  W.     Design  of  a  Condensing  Plant 12 mo,  *i  50 

Wright,  T.  W.     Elements  of  Mechanics 8vo,  *2  50 

Wright,  T.  W.,  and  Hayford,  J.  F.     Adjustment  of  Observations 

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Zeidler,  J.,  and  Lustgarten,  J.     Electric  Arc  Lamps 8vo,  *2  00 

Zeuner,    A.     Technical     Thermodynamics.     Trans,    by    J.    F. 

Klein.     Two  Volumes Bvo,  *B  00 

Zimmer,  G.  F.     Mechanical  Handling  of  Material 4to,  *io  00 

Zipser,  J.     Textile  Raw  Materials.     Trans,  by  C.  Salter Bvo,  *5  00 

Zur   Nedden,  F.     Engineering  Workshop  Machines  and  Proc- 

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